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跳转至
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Hello 算法
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7.1. 二叉树
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0. 写在前面
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0. 写在前面
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0.1. 关于本书
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0.2. 如何使用本书
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0.3. 小结
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1. 引言
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1. 引言
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1.1. 算法无处不在
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1.2. 算法是什么
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1.3. 小结
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2. 复杂度分析
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2. 复杂度分析
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2.1. 算法效率评估
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< a href = "../../chapter_computational_complexity/time_complexity/" class = "md-nav__link" >
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2.2. 时间复杂度
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2.3. 空间复杂度
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< a href = "../../chapter_computational_complexity/space_time_tradeoff/" class = "md-nav__link" >
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2.4. 权衡时间与空间
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< a href = "../../chapter_computational_complexity/summary/" class = "md-nav__link" >
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2.5. 小结
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3. 数据结构简介
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3. 数据结构简介
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3.1. 数据与内存
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< a href = "../../chapter_data_structure/classification_of_data_structure/" class = "md-nav__link" >
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3.2. 数据结构分类
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3.3. 小结
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4. 数组与链表
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< span class = "md-nav__icon md-icon" > < / span >
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4. 数组与链表
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< a href = "../../chapter_array_and_linkedlist/array/" class = "md-nav__link" >
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4.1. 数组
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< a href = "../../chapter_array_and_linkedlist/linked_list/" class = "md-nav__link" >
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4.2. 链表
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< a href = "../../chapter_array_and_linkedlist/list/" class = "md-nav__link" >
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4.3. 列表
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4.4. 小结
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5. 栈与队列
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5. 栈与队列
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< a href = "../../chapter_stack_and_queue/stack/" class = "md-nav__link" >
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5.1. 栈
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< a href = "../../chapter_stack_and_queue/queue/" class = "md-nav__link" >
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5.2. 队列
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< a href = "../../chapter_stack_and_queue/deque/" class = "md-nav__link" >
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5.3. 双向队列
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5.4. 小结
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6. 散列表
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6. 散列表
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< a href = "../../chapter_hashing/hash_map/" class = "md-nav__link" >
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6.1. 哈希表
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< a href = "../../chapter_hashing/hash_collision/" class = "md-nav__link" >
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6.2. 哈希冲突处理
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6.3. 小结
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< label class = "md-nav__link" for = "__nav_8" id = "__nav_8_label" tabindex = "0" >
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7. 树
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7. 树
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7.1. 二叉树
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7.1. 二叉树
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目录
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7.1.1. 二叉树常见术语
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7.1.2. 二叉树基本操作
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7.1.3. 常见二叉树类型
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完美二叉树
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完全二叉树
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完满二叉树
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平衡二叉树
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7.1.4. 二叉树的退化
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7.1.5. 二叉树表示方式 *
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7.2. 二叉树遍历
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< a href = "../binary_search_tree/" class = "md-nav__link" >
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7.3. 二叉搜索树
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< a href = "../avl_tree/" class = "md-nav__link" >
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7.4. AVL 树 *
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7.5. 小结
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< label class = "md-nav__link" for = "__nav_9" id = "__nav_9_label" tabindex = "0" >
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8. 堆
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8. 堆
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8.1. 堆
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8.2. 建堆操作 *
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8.3. 小结
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< label class = "md-nav__link" for = "__nav_10" id = "__nav_10_label" tabindex = "0" >
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9. 图
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9. 图
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9.1. 图
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< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
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9.2. 图基础操作
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< a href = "../../chapter_graph/graph_traversal/" class = "md-nav__link" >
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9.3. 图的遍历（ ）
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< a href = "../../chapter_graph/summary/" class = "md-nav__link" >
9.4. 小结
< / a >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_11" >
< label class = "md-nav__link" for = "__nav_11" id = "__nav_11_label" tabindex = "0" >
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10. 查找算法
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< label class = "md-nav__title" for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
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10. 查找算法
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/linear_search/" class = "md-nav__link" >
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10.1. 线性查找
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< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
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10.2. 二分查找
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< a href = "../../chapter_searching/hashing_search/" class = "md-nav__link" >
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10.3. 哈希查找
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< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
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10.4. 小结
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_12" >
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< label class = "md-nav__link" for = "__nav_12" id = "__nav_12_label" tabindex = "0" >
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11. 排序算法
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< label class = "md-nav__title" for = "__nav_12" >
< span class = "md-nav__icon md-icon" > < / span >
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11. 排序算法
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< a href = "../../chapter_sorting/intro_to_sort/" class = "md-nav__link" >
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11.1. 排序简介
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< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
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11.2. 冒泡排序
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< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/insertion_sort/" class = "md-nav__link" >
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11.3. 插入排序
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< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/quick_sort/" class = "md-nav__link" >
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11.4. 快速排序
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< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/merge_sort/" class = "md-nav__link" >
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11.5. 归并排序
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/counting_sort/" class = "md-nav__link" >
11.6. 计数排序（ ）
< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/summary/" class = "md-nav__link" >
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11.7. 小结
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< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_13" >
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< label class = "md-nav__link" for = "__nav_13" id = "__nav_13_label" tabindex = "0" >
12. 附录
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_13_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_13" >
< span class = "md-nav__icon md-icon" > < / span >
12. 附录
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_appendix/installation/" class = "md-nav__link" >
12.1. 编程环境安装
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_appendix/contribution/" class = "md-nav__link" >
12.2. 一起参与创作
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_14" >
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< div class = "md-nav__link md-nav__link--index " >
< a href = "../../chapter_reference/" > 参考文献< / a >
< / div >
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< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_14_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_14" >
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< span class = "md-nav__icon md-icon" > < / span >
参考文献
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< / ul >
< / nav >
< / li >
< / ul >
< / nav >
< / div >
< / div >
< / div >
< div class = "md-sidebar md-sidebar--secondary" data-md-component = "sidebar" data-md-type = "toc" >
< div class = "md-sidebar__scrollwrap" >
< div class = "md-sidebar__inner" >
< nav class = "md-nav md-nav--secondary" aria-label = "目录" >
< label class = "md-nav__title" for = "__toc" >
< span class = "md-nav__icon md-icon" > < / span >
目录
< / label >
< ul class = "md-nav__list" data-md-component = "toc" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "#711" class = "md-nav__link" >
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7.1.1. 二叉树常见术语
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< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#712" class = "md-nav__link" >
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7.1.2. 二叉树基本操作
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< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#713" class = "md-nav__link" >
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7.1.3. 常见二叉树类型
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< / a >
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< nav class = "md-nav" aria-label = "7.1.3. 常见二叉树类型" >
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< ul class = "md-nav__list" >
< li class = "md-nav__item" >
< a href = "#_1" class = "md-nav__link" >
完美二叉树
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_2" class = "md-nav__link" >
完全二叉树
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_3" class = "md-nav__link" >
完满二叉树
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_4" class = "md-nav__link" >
平衡二叉树
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item" >
< a href = "#714" class = "md-nav__link" >
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7.1.4. 二叉树的退化
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< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#715" class = "md-nav__link" >
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7.1.5. 二叉树表示方式 *
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< / a >
< / li >
< / ul >
< / nav >
< / div >
< / div >
< / div >
< div class = "md-content" data-md-component = "content" >
< article class = "md-content__inner md-typeset" >
< a href = "https://github.com/krahets/hello-algo/tree/main/docs/chapter_tree/binary_tree.md" title = "编辑此页" class = "md-content__button md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z" / > < / svg >
< / a >
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< h1 id = "71" > 7.1. 二叉树< a class = "headerlink" href = "#71" title = "Permanent link" > ¶ < / a > < / h1 >
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< p > 「二叉树 Binary Tree」是一种非线性数据结构， ， ， ， < / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "1:10" > < input checked = "checked" id = "__tabbed_1_1" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_2" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_3" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_4" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_5" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_6" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_7" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_8" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_9" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_10" name = "__tabbed_1" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_1_1" > Java< / label > < label for = "__tabbed_1_2" > C++< / label > < label for = "__tabbed_1_3" > Python< / label > < label for = "__tabbed_1_4" > Go< / label > < label for = "__tabbed_1_5" > JavaScript< / label > < label for = "__tabbed_1_6" > TypeScript< / label > < label for = "__tabbed_1_7" > C< / label > < label for = "__tabbed_1_8" > C#< / label > < label for = "__tabbed_1_9" > Swift< / label > < label for = "__tabbed_1_10" > Zig< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "cm" > /* 二叉树结点类 */< / span >
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< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "kd" > class< / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 结点值< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 左子结点指针< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 右子结点指针< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-1-1" name = "__codelineno-1-1" href = "#__codelineno-1-1" > < / a > < span class = "cm" > /* 二叉树结点结构体 */< / span >
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< a id = "__codelineno-1-2" name = "__codelineno-1-2" href = "#__codelineno-1-2" > < / a > < span class = "k" > struct< / span > < span class = "w" > < / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-1-3" name = "__codelineno-1-3" href = "#__codelineno-1-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 结点值< / span >
< a id = "__codelineno-1-4" name = "__codelineno-1-4" href = "#__codelineno-1-4" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 左子结点指针< / span >
< a id = "__codelineno-1-5" name = "__codelineno-1-5" href = "#__codelineno-1-5" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "n" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 右子结点指针< / span >
< a id = "__codelineno-1-6" name = "__codelineno-1-6" href = "#__codelineno-1-6" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > ),< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "p" > (< / span > < span class = "k" > nullptr< / span > < span class = "p" > ),< / span > < span class = "w" > < / span > < span class = "n" > right< / span > < span class = "p" > (< / span > < span class = "k" > nullptr< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {}< / span >
< a id = "__codelineno-1-7" name = "__codelineno-1-7" href = "#__codelineno-1-7" > < / a > < span class = "p" > };< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "sd" > " " " 二叉树结点类 " " " < / span >
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< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a > < span class = "k" > class< / span > < span class = "nc" > TreeNode< / span > < span class = "p" > :< / span >
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< a id = "__codelineno-2-3" name = "__codelineno-2-3" href = "#__codelineno-2-3" > < / a > < span class = "k" > def< / span > < span class = "fm" > __init__< / span > < span class = "p" > (< / span > < span class = "bp" > self< / span > < span class = "p" > ,< / span > < span class = "n" > val< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-2-4" name = "__codelineno-2-4" href = "#__codelineno-2-4" > < / a > < span class = "bp" > self< / span > < span class = "o" > .< / span > < span class = "n" > val< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "o" > =< / span > < span class = "n" > val< / span > < span class = "c1" > # 结点值< / span >
< a id = "__codelineno-2-5" name = "__codelineno-2-5" href = "#__codelineno-2-5" > < / a > < span class = "bp" > self< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "p" > :< / span > < span class = "n" > Optional< / span > < span class = "p" > [< / span > < span class = "n" > TreeNode< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "c1" > # 左子结点指针< / span >
< a id = "__codelineno-2-6" name = "__codelineno-2-6" href = "#__codelineno-2-6" > < / a > < span class = "bp" > self< / span > < span class = "o" > .< / span > < span class = "n" > right< / span > < span class = "p" > :< / span > < span class = "n" > Optional< / span > < span class = "p" > [< / span > < span class = "n" > TreeNode< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "kc" > None< / span > < span class = "c1" > # 右子结点指针< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-3-1" name = "__codelineno-3-1" href = "#__codelineno-3-1" > < / a > < span class = "cm" > /* 二叉树结点结构体 */< / span >
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< a id = "__codelineno-3-2" name = "__codelineno-3-2" href = "#__codelineno-3-2" > < / a > < span class = "kd" > type< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "w" > < / span > < span class = "kd" > struct< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-3-3" name = "__codelineno-3-3" href = "#__codelineno-3-3" > < / a > < span class = "w" > < / span > < span class = "nx" > Val< / span > < span class = "w" > < / span > < span class = "kt" > int< / span >
< a id = "__codelineno-3-4" name = "__codelineno-3-4" href = "#__codelineno-3-4" > < / a > < span class = "w" > < / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "nx" > TreeNode< / span >
< a id = "__codelineno-3-5" name = "__codelineno-3-5" href = "#__codelineno-3-5" > < / a > < span class = "w" > < / span > < span class = "nx" > Right< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "nx" > TreeNode< / span >
< a id = "__codelineno-3-6" name = "__codelineno-3-6" href = "#__codelineno-3-6" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-3-7" name = "__codelineno-3-7" href = "#__codelineno-3-7" > < / a > < span class = "cm" > /* 结点初始化方法 */< / span >
< a id = "__codelineno-3-8" name = "__codelineno-3-8" href = "#__codelineno-3-8" > < / a > < span class = "kd" > func< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "nx" > v< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "nx" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-3-9" name = "__codelineno-3-9" href = "#__codelineno-3-9" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > {< / span >
< a id = "__codelineno-3-10" name = "__codelineno-3-10" href = "#__codelineno-3-10" > < / a > < span class = "w" > < / span > < span class = "nx" > Left< / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-3-11" name = "__codelineno-3-11" href = "#__codelineno-3-11" > < / a > < span class = "w" > < / span > < span class = "nx" > Right< / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-3-12" name = "__codelineno-3-12" href = "#__codelineno-3-12" > < / a > < span class = "w" > < / span > < span class = "nx" > Val< / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "nx" > v< / span > < span class = "p" > ,< / span >
< a id = "__codelineno-3-13" name = "__codelineno-3-13" href = "#__codelineno-3-13" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-3-14" name = "__codelineno-3-14" href = "#__codelineno-3-14" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-4-1" name = "__codelineno-4-1" href = "#__codelineno-4-1" > < / a > < span class = "cm" > /* 二叉树结点类 */< / span >
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< a id = "__codelineno-4-2" name = "__codelineno-4-2" href = "#__codelineno-4-2" > < / a > < span class = "kd" > function< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "nx" > val< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-4-3" name = "__codelineno-4-3" href = "#__codelineno-4-3" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > val< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "nx" > val< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "mf" > 0< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "nx" > val< / span > < span class = "p" > );< / span > < span class = "w" > < / span > < span class = "c1" > // 结点值< / span >
< a id = "__codelineno-4-4" name = "__codelineno-4-4" href = "#__codelineno-4-4" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "p" > );< / span > < span class = "w" > < / span > < span class = "c1" > // 左子结点指针< / span >
< a id = "__codelineno-4-5" name = "__codelineno-4-5" href = "#__codelineno-4-5" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "p" > );< / span > < span class = "w" > < / span > < span class = "c1" > // 右子结点指针< / span >
< a id = "__codelineno-4-6" name = "__codelineno-4-6" href = "#__codelineno-4-6" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-5-1" name = "__codelineno-5-1" href = "#__codelineno-5-1" > < / a > < span class = "cm" > /* 二叉树结点类 */< / span >
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< a id = "__codelineno-5-2" name = "__codelineno-5-2" href = "#__codelineno-5-2" > < / a > < span class = "kd" > class< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-5-3" name = "__codelineno-5-3" href = "#__codelineno-5-3" > < / a > < span class = "w" > < / span > < span class = "nx" > val< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > number< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-5-4" name = "__codelineno-5-4" href = "#__codelineno-5-4" > < / a > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-5-5" name = "__codelineno-5-5" href = "#__codelineno-5-5" > < / a > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-5-6" name = "__codelineno-5-6" href = "#__codelineno-5-6" > < / a >
< a id = "__codelineno-5-7" name = "__codelineno-5-7" href = "#__codelineno-5-7" > < / a > < span class = "w" > < / span > < span class = "kr" > constructor< / span > < span class = "p" > (< / span > < span class = "nx" > val?< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > number< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > left?< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "nx" > right?< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > TreeNode< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-5-8" name = "__codelineno-5-8" href = "#__codelineno-5-8" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > val< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > val< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "nx" > 0< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 结点值< / span >
< a id = "__codelineno-5-9" name = "__codelineno-5-9" href = "#__codelineno-5-9" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "nx" > null< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 左子结点指针< / span >
< a id = "__codelineno-5-10" name = "__codelineno-5-10" href = "#__codelineno-5-10" > < / a > < span class = "w" > < / span > < span class = "k" > this< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > ===< / span > < span class = "w" > < / span > < span class = "kc" > undefined< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "nx" > null< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "kt" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 右子结点指针< / span >
< a id = "__codelineno-5-11" name = "__codelineno-5-11" href = "#__codelineno-5-11" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-5-12" name = "__codelineno-5-12" href = "#__codelineno-5-12" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-6-1" name = "__codelineno-6-1" href = "#__codelineno-6-1" > < / a >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-7-1" name = "__codelineno-7-1" href = "#__codelineno-7-1" > < / a > < span class = "cm" > /* 二叉树结点类 */< / span >
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< a id = "__codelineno-7-2" name = "__codelineno-7-2" href = "#__codelineno-7-2" > < / a > < span class = "k" > class< / span > < span class = "w" > < / span > < span class = "nc" > TreeNode< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-7-3" name = "__codelineno-7-3" href = "#__codelineno-7-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 结点值< / span >
< a id = "__codelineno-7-4" name = "__codelineno-7-4" href = "#__codelineno-7-4" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 左子结点指针< / span >
< a id = "__codelineno-7-5" name = "__codelineno-7-5" href = "#__codelineno-7-5" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "n" > right< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "c1" > // 右子结点指针< / span >
< a id = "__codelineno-7-6" name = "__codelineno-7-6" href = "#__codelineno-7-6" > < / a > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "n" > val< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > x< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-7-7" name = "__codelineno-7-7" href = "#__codelineno-7-7" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-8-1" name = "__codelineno-8-1" href = "#__codelineno-8-1" > < / a > < span class = "cm" > /* 二叉树结点类 */< / span >
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< a id = "__codelineno-8-2" name = "__codelineno-8-2" href = "#__codelineno-8-2" > < / a > < span class = "kd" > class< / span > < span class = "nc" > TreeNode< / span > < span class = "p" > {< / span >
< a id = "__codelineno-8-3" name = "__codelineno-8-3" href = "#__codelineno-8-3" > < / a > < span class = "kd" > var< / span > < span class = "nv" > val< / span > < span class = "p" > :< / span > < span class = "nb" > Int< / span > < span class = "c1" > // 结点值< / span >
< a id = "__codelineno-8-4" name = "__codelineno-8-4" href = "#__codelineno-8-4" > < / a > < span class = "kd" > var< / span > < span class = "nv" > left< / span > < span class = "p" > :< / span > < span class = "n" > TreeNode< / span > < span class = "p" > ?< / span > < span class = "c1" > // 左子结点指针< / span >
< a id = "__codelineno-8-5" name = "__codelineno-8-5" href = "#__codelineno-8-5" > < / a > < span class = "kd" > var< / span > < span class = "nv" > right< / span > < span class = "p" > :< / span > < span class = "n" > TreeNode< / span > < span class = "p" > ?< / span > < span class = "c1" > // 右子结点指针< / span >
< a id = "__codelineno-8-6" name = "__codelineno-8-6" href = "#__codelineno-8-6" > < / a >
< a id = "__codelineno-8-7" name = "__codelineno-8-7" href = "#__codelineno-8-7" > < / a > < span class = "kd" > init< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "nb" > Int< / span > < span class = "p" > )< / span > < span class = "p" > {< / span >
< a id = "__codelineno-8-8" name = "__codelineno-8-8" href = "#__codelineno-8-8" > < / a > < span class = "n" > val< / span > < span class = "p" > =< / span > < span class = "n" > x< / span >
< a id = "__codelineno-8-9" name = "__codelineno-8-9" href = "#__codelineno-8-9" > < / a > < span class = "p" > }< / span >
< a id = "__codelineno-8-10" name = "__codelineno-8-10" href = "#__codelineno-8-10" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-9-1" name = "__codelineno-9-1" href = "#__codelineno-9-1" > < / a >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< p > 结点的两个指针分别指向「左子结点 Left Child Node」和「右子结点 Right Child Node」， ， ， < / p >
< p > 除了叶结点外，每个结点都有子结点和子树。例如，若将下图的「结点 2」看作父结点， ， < / p >
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< p > < img alt = "父结点、子结点、子树" src = "../binary_tree.assets/binary_tree_definition.png" / > < / p >
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< p align = "center" > Fig. 父结点、子结点、子树 < / p >
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< h2 id = "711" > 7.1.1. 二叉树常见术语< a class = "headerlink" href = "#711" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 二叉树的术语较多，建议尽量理解并记住。后续可能遗忘，可以在需要使用时回来查看确认。< / p >
< ul >
< li > 「根结点 Root Node」： ， ； < / li >
< li > 「叶结点 Leaf Node」： ， < span class = "arithmatex" > \(\text{null}\)< / span > ； < / li >
< li > 结点所处「层 Level」： ， ； < / li >
< li > 结点「度 Degree」： ， ； < / li >
< li > 「边 Edge」： ， ； < / li >
< li > 二叉树「高度」：二叉树中根结点到最远叶结点走过边的数量；< / li >
< li > 结点「深度 Depth」 ：根结点到该结点走过边的数量；< / li >
< li > 结点「高度 Height」： ； < / li >
< / ul >
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< p > < img alt = "二叉树的常用术语" src = "../binary_tree.assets/binary_tree_terminology.png" / > < / p >
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< p align = "center" > Fig. 二叉树的常用术语 < / p >
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< div class = "admonition tip" >
< p class = "admonition-title" > 高度与深度的定义< / p >
< p > 值得注意，我们通常将「高度」和「深度」定义为“走过边的数量”，而有些题目或教材会将其定义为“走过结点的数量”，此时高度或深度都需要 + 1 。< / p >
< / div >
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< h2 id = "712" > 7.1.2. 二叉树基本操作< a class = "headerlink" href = "#712" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > < strong > 初始化二叉树< / strong > 。与链表类似，先初始化结点，再构建引用指向（即指针）。< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "2:10" > < input checked = "checked" id = "__tabbed_2_1" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_2" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_3" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_4" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_5" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_6" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_7" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_8" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_9" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_10" name = "__tabbed_2" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_2_1" > Java< / label > < label for = "__tabbed_2_2" > C++< / label > < label for = "__tabbed_2_3" > Python< / label > < label for = "__tabbed_2_4" > Go< / label > < label for = "__tabbed_2_5" > JavaScript< / label > < label for = "__tabbed_2_6" > TypeScript< / label > < label for = "__tabbed_2_7" > C< / label > < label for = "__tabbed_2_8" > C#< / label > < label for = "__tabbed_2_9" > Swift< / label > < label for = "__tabbed_2_10" > Zig< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-10-1" name = "__codelineno-10-1" href = "#__codelineno-10-1" > < / a > < span class = "c1" > // 初始化结点< / span >
< a id = "__codelineno-10-2" name = "__codelineno-10-2" href = "#__codelineno-10-2" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-10-3" name = "__codelineno-10-3" href = "#__codelineno-10-3" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 2< / span > < span class = "p" > );< / span >
< a id = "__codelineno-10-4" name = "__codelineno-10-4" href = "#__codelineno-10-4" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-10-5" name = "__codelineno-10-5" href = "#__codelineno-10-5" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-10-6" name = "__codelineno-10-6" href = "#__codelineno-10-6" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-10-7" name = "__codelineno-10-7" href = "#__codelineno-10-7" > < / a > < span class = "c1" > // 构建引用指向（即指针）< / span >
< a id = "__codelineno-10-8" name = "__codelineno-10-8" href = "#__codelineno-10-8" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-9" name = "__codelineno-10-9" href = "#__codelineno-10-9" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-10" name = "__codelineno-10-10" href = "#__codelineno-10-10" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-10-11" name = "__codelineno-10-11" href = "#__codelineno-10-11" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "na" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-11-1" name = "__codelineno-11-1" href = "#__codelineno-11-1" > < / a > < span class = "cm" > /* 初始化二叉树 */< / span >
< a id = "__codelineno-11-2" name = "__codelineno-11-2" href = "#__codelineno-11-2" > < / a > < span class = "c1" > // 初始化结点< / span >
< a id = "__codelineno-11-3" name = "__codelineno-11-3" href = "#__codelineno-11-3" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-11-4" name = "__codelineno-11-4" href = "#__codelineno-11-4" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 2< / span > < span class = "p" > );< / span >
< a id = "__codelineno-11-5" name = "__codelineno-11-5" href = "#__codelineno-11-5" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-11-6" name = "__codelineno-11-6" href = "#__codelineno-11-6" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-11-7" name = "__codelineno-11-7" href = "#__codelineno-11-7" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-11-8" name = "__codelineno-11-8" href = "#__codelineno-11-8" > < / a > < span class = "c1" > // 构建引用指向（即指针）< / span >
< a id = "__codelineno-11-9" name = "__codelineno-11-9" href = "#__codelineno-11-9" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-11-10" name = "__codelineno-11-10" href = "#__codelineno-11-10" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-11-11" name = "__codelineno-11-11" href = "#__codelineno-11-11" > < / a > < span class = "n" > n2< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-11-12" name = "__codelineno-11-12" href = "#__codelineno-11-12" > < / a > < span class = "n" > n2< / span > < span class = "o" > -> < / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-12-1" name = "__codelineno-12-1" href = "#__codelineno-12-1" > < / a > < span class = "sd" > " " " 初始化二叉树 " " " < / span >
< a id = "__codelineno-12-2" name = "__codelineno-12-2" href = "#__codelineno-12-2" > < / a > < span class = "c1" > # 初始化结点< / span >
< a id = "__codelineno-12-3" name = "__codelineno-12-3" href = "#__codelineno-12-3" > < / a > < span class = "n" > n1< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-12-4" name = "__codelineno-12-4" href = "#__codelineno-12-4" > < / a > < span class = "n" > n2< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-12-5" name = "__codelineno-12-5" href = "#__codelineno-12-5" > < / a > < span class = "n" > n3< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 3< / span > < span class = "p" > )< / span >
< a id = "__codelineno-12-6" name = "__codelineno-12-6" href = "#__codelineno-12-6" > < / a > < span class = "n" > n4< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 4< / span > < span class = "p" > )< / span >
< a id = "__codelineno-12-7" name = "__codelineno-12-7" href = "#__codelineno-12-7" > < / a > < span class = "n" > n5< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > val< / span > < span class = "o" > =< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-12-8" name = "__codelineno-12-8" href = "#__codelineno-12-8" > < / a > < span class = "c1" > # 构建引用指向（即指针）< / span >
< a id = "__codelineno-12-9" name = "__codelineno-12-9" href = "#__codelineno-12-9" > < / a > < span class = "n" > n1< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > n2< / span >
< a id = "__codelineno-12-10" name = "__codelineno-12-10" href = "#__codelineno-12-10" > < / a > < span class = "n" > n1< / span > < span class = "o" > .< / span > < span class = "n" > right< / span > < span class = "o" > =< / span > < span class = "n" > n3< / span >
< a id = "__codelineno-12-11" name = "__codelineno-12-11" href = "#__codelineno-12-11" > < / a > < span class = "n" > n2< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > n4< / span >
< a id = "__codelineno-12-12" name = "__codelineno-12-12" href = "#__codelineno-12-12" > < / a > < span class = "n" > n2< / span > < span class = "o" > .< / span > < span class = "n" > right< / span > < span class = "o" > =< / span > < span class = "n" > n5< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-13-1" name = "__codelineno-13-1" href = "#__codelineno-13-1" > < / a > < span class = "cm" > /* 初始化二叉树 */< / span >
< a id = "__codelineno-13-2" name = "__codelineno-13-2" href = "#__codelineno-13-2" > < / a > < span class = "c1" > // 初始化结点< / span >
< a id = "__codelineno-13-3" name = "__codelineno-13-3" href = "#__codelineno-13-3" > < / a > < span class = "nx" > n1< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-13-4" name = "__codelineno-13-4" href = "#__codelineno-13-4" > < / a > < span class = "nx" > n2< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-13-5" name = "__codelineno-13-5" href = "#__codelineno-13-5" > < / a > < span class = "nx" > n3< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > )< / span >
< a id = "__codelineno-13-6" name = "__codelineno-13-6" href = "#__codelineno-13-6" > < / a > < span class = "nx" > n4< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 4< / span > < span class = "p" > )< / span >
< a id = "__codelineno-13-7" name = "__codelineno-13-7" href = "#__codelineno-13-7" > < / a > < span class = "nx" > n5< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-13-8" name = "__codelineno-13-8" href = "#__codelineno-13-8" > < / a > < span class = "c1" > // 构建引用指向（即指针）< / span >
< a id = "__codelineno-13-9" name = "__codelineno-13-9" href = "#__codelineno-13-9" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span >
< a id = "__codelineno-13-10" name = "__codelineno-13-10" href = "#__codelineno-13-10" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > Right< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n3< / span >
< a id = "__codelineno-13-11" name = "__codelineno-13-11" href = "#__codelineno-13-11" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n4< / span >
< a id = "__codelineno-13-12" name = "__codelineno-13-12" href = "#__codelineno-13-12" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > Right< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n5< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-14-1" name = "__codelineno-14-1" href = "#__codelineno-14-1" > < / a > < span class = "cm" > /* 初始化二叉树 */< / span >
< a id = "__codelineno-14-2" name = "__codelineno-14-2" href = "#__codelineno-14-2" > < / a > < span class = "c1" > // 初始化结点< / span >
< a id = "__codelineno-14-3" name = "__codelineno-14-3" href = "#__codelineno-14-3" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 1< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-14-4" name = "__codelineno-14-4" href = "#__codelineno-14-4" > < / a > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 2< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-14-5" name = "__codelineno-14-5" href = "#__codelineno-14-5" > < / a > < span class = "w" > < / span > < span class = "nx" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 3< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-14-6" name = "__codelineno-14-6" href = "#__codelineno-14-6" > < / a > < span class = "w" > < / span > < span class = "nx" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 4< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-14-7" name = "__codelineno-14-7" href = "#__codelineno-14-7" > < / a > < span class = "w" > < / span > < span class = "nx" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-14-8" name = "__codelineno-14-8" href = "#__codelineno-14-8" > < / a > < span class = "c1" > // 构建引用指向（即指针）< / span >
< a id = "__codelineno-14-9" name = "__codelineno-14-9" href = "#__codelineno-14-9" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-14-10" name = "__codelineno-14-10" href = "#__codelineno-14-10" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-14-11" name = "__codelineno-14-11" href = "#__codelineno-14-11" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-14-12" name = "__codelineno-14-12" href = "#__codelineno-14-12" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-15-1" name = "__codelineno-15-1" href = "#__codelineno-15-1" > < / a > < span class = "cm" > /* 初始化二叉树 */< / span >
< a id = "__codelineno-15-2" name = "__codelineno-15-2" href = "#__codelineno-15-2" > < / a > < span class = "c1" > // 初始化结点< / span >
< a id = "__codelineno-15-3" name = "__codelineno-15-3" href = "#__codelineno-15-3" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 1< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-15-4" name = "__codelineno-15-4" href = "#__codelineno-15-4" > < / a > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 2< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-15-5" name = "__codelineno-15-5" href = "#__codelineno-15-5" > < / a > < span class = "w" > < / span > < span class = "nx" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 3< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-15-6" name = "__codelineno-15-6" href = "#__codelineno-15-6" > < / a > < span class = "w" > < / span > < span class = "nx" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 4< / span > < span class = "p" > ),< / span >
< a id = "__codelineno-15-7" name = "__codelineno-15-7" href = "#__codelineno-15-7" > < / a > < span class = "w" > < / span > < span class = "nx" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-15-8" name = "__codelineno-15-8" href = "#__codelineno-15-8" > < / a > < span class = "c1" > // 构建引用指向（即指针）< / span >
< a id = "__codelineno-15-9" name = "__codelineno-15-9" href = "#__codelineno-15-9" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-15-10" name = "__codelineno-15-10" href = "#__codelineno-15-10" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-15-11" name = "__codelineno-15-11" href = "#__codelineno-15-11" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-15-12" name = "__codelineno-15-12" href = "#__codelineno-15-12" > < / a > < span class = "nx" > n2< / span > < span class = "p" > .< / span > < span class = "nx" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-16-1" name = "__codelineno-16-1" href = "#__codelineno-16-1" > < / a >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-17-1" name = "__codelineno-17-1" href = "#__codelineno-17-1" > < / a > < span class = "cm" > /* 初始化二叉树 */< / span >
< a id = "__codelineno-17-2" name = "__codelineno-17-2" href = "#__codelineno-17-2" > < / a > < span class = "c1" > // 初始化结点< / span >
< a id = "__codelineno-17-3" name = "__codelineno-17-3" href = "#__codelineno-17-3" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n1< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-4" name = "__codelineno-17-4" href = "#__codelineno-17-4" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 2< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-5" name = "__codelineno-17-5" href = "#__codelineno-17-5" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 3< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-6" name = "__codelineno-17-6" href = "#__codelineno-17-6" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 4< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-7" name = "__codelineno-17-7" href = "#__codelineno-17-7" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 5< / span > < span class = "p" > );< / span >
< a id = "__codelineno-17-8" name = "__codelineno-17-8" href = "#__codelineno-17-8" > < / a > < span class = "c1" > // 构建引用指向（即指针）< / span >
< a id = "__codelineno-17-9" name = "__codelineno-17-9" href = "#__codelineno-17-9" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-17-10" name = "__codelineno-17-10" href = "#__codelineno-17-10" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n3< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-17-11" name = "__codelineno-17-11" href = "#__codelineno-17-11" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n4< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-17-12" name = "__codelineno-17-12" href = "#__codelineno-17-12" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "n" > right< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n5< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-18-1" name = "__codelineno-18-1" href = "#__codelineno-18-1" > < / a > < span class = "c1" > // 初始化结点< / span >
< a id = "__codelineno-18-2" name = "__codelineno-18-2" href = "#__codelineno-18-2" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n1< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-3" name = "__codelineno-18-3" href = "#__codelineno-18-3" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n2< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-4" name = "__codelineno-18-4" href = "#__codelineno-18-4" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n3< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 3< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-5" name = "__codelineno-18-5" href = "#__codelineno-18-5" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n4< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 4< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-6" name = "__codelineno-18-6" href = "#__codelineno-18-6" > < / a > < span class = "kd" > let< / span > < span class = "nv" > n5< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 5< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-7" name = "__codelineno-18-7" href = "#__codelineno-18-7" > < / a > < span class = "c1" > // 构建引用指向（即指针）< / span >
< a id = "__codelineno-18-8" name = "__codelineno-18-8" href = "#__codelineno-18-8" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > n2< / span >
< a id = "__codelineno-18-9" name = "__codelineno-18-9" href = "#__codelineno-18-9" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "kr" > right< / span > < span class = "p" > =< / span > < span class = "n" > n3< / span >
< a id = "__codelineno-18-10" name = "__codelineno-18-10" href = "#__codelineno-18-10" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > n4< / span >
< a id = "__codelineno-18-11" name = "__codelineno-18-11" href = "#__codelineno-18-11" > < / a > < span class = "n" > n2< / span > < span class = "p" > .< / span > < span class = "kr" > right< / span > < span class = "p" > =< / span > < span class = "n" > n5< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-19-1" name = "__codelineno-19-1" href = "#__codelineno-19-1" > < / a >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< p > < strong > 插入与删除结点< / strong > 。与链表类似，插入与删除结点都可以通过修改指针实现。< / p >
2023-02-26 18:18:03 +08:00
< p > < img alt = "在二叉树中插入与删除结点" src = "../binary_tree.assets/binary_tree_add_remove.png" / > < / p >
2023-02-26 19:53:32 +08:00
< p align = "center" > Fig. 在二叉树中插入与删除结点 < / p >
2023-02-08 14:54:41 +08:00
< div class = "tabbed-set tabbed-alternate" data-tabs = "3:10" > < input checked = "checked" id = "__tabbed_3_1" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_2" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_3" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_4" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_5" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_6" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_7" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_8" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_9" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_10" name = "__tabbed_3" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_3_1" > Java< / label > < label for = "__tabbed_3_2" > C++< / label > < label for = "__tabbed_3_3" > Python< / label > < label for = "__tabbed_3_4" > Go< / label > < label for = "__tabbed_3_5" > JavaScript< / label > < label for = "__tabbed_3_6" > TypeScript< / label > < label for = "__tabbed_3_7" > C< / label > < label for = "__tabbed_3_8" > C#< / label > < label for = "__tabbed_3_9" > Swift< / label > < label for = "__tabbed_3_10" > Zig< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-20-1" name = "__codelineno-20-1" href = "#__codelineno-20-1" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-20-2" name = "__codelineno-20-2" href = "#__codelineno-20-2" > < / a > < span class = "c1" > // 在 n1 -> n2 中间插入结点 P< / span >
< a id = "__codelineno-20-3" name = "__codelineno-20-3" href = "#__codelineno-20-3" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-20-4" name = "__codelineno-20-4" href = "#__codelineno-20-4" > < / a > < span class = "n" > P< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-20-5" name = "__codelineno-20-5" href = "#__codelineno-20-5" > < / a > < span class = "c1" > // 删除结点 P< / span >
< a id = "__codelineno-20-6" name = "__codelineno-20-6" href = "#__codelineno-20-6" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "na" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-21-1" name = "__codelineno-21-1" href = "#__codelineno-21-1" > < / a > < span class = "cm" > /* 插入与删除结点 */< / span >
< a id = "__codelineno-21-2" name = "__codelineno-21-2" href = "#__codelineno-21-2" > < / a > < span class = "n" > TreeNode< / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-21-3" name = "__codelineno-21-3" href = "#__codelineno-21-3" > < / a > < span class = "c1" > // 在 n1 -> n2 中间插入结点 P< / span >
< a id = "__codelineno-21-4" name = "__codelineno-21-4" href = "#__codelineno-21-4" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-21-5" name = "__codelineno-21-5" href = "#__codelineno-21-5" > < / a > < span class = "n" > P< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-21-6" name = "__codelineno-21-6" href = "#__codelineno-21-6" > < / a > < span class = "c1" > // 删除结点 P< / span >
< a id = "__codelineno-21-7" name = "__codelineno-21-7" href = "#__codelineno-21-7" > < / a > < span class = "n" > n1< / span > < span class = "o" > -> < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-22-1" name = "__codelineno-22-1" href = "#__codelineno-22-1" > < / a > < span class = "sd" > " " " 插入与删除结点 " " " < / span >
< a id = "__codelineno-22-2" name = "__codelineno-22-2" href = "#__codelineno-22-2" > < / a > < span class = "n" > p< / span > < span class = "o" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span >
< a id = "__codelineno-22-3" name = "__codelineno-22-3" href = "#__codelineno-22-3" > < / a > < span class = "c1" > # 在 n1 -> n2 中间插入结点 P< / span >
< a id = "__codelineno-22-4" name = "__codelineno-22-4" href = "#__codelineno-22-4" > < / a > < span class = "n" > n1< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > p< / span >
< a id = "__codelineno-22-5" name = "__codelineno-22-5" href = "#__codelineno-22-5" > < / a > < span class = "n" > p< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > n2< / span >
< a id = "__codelineno-22-6" name = "__codelineno-22-6" href = "#__codelineno-22-6" > < / a > < span class = "c1" > # 删除结点 P< / span >
< a id = "__codelineno-22-7" name = "__codelineno-22-7" href = "#__codelineno-22-7" > < / a > < span class = "n" > n1< / span > < span class = "o" > .< / span > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > n2< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-23-1" name = "__codelineno-23-1" href = "#__codelineno-23-1" > < / a > < span class = "cm" > /* 插入与删除结点 */< / span >
< a id = "__codelineno-23-2" name = "__codelineno-23-2" href = "#__codelineno-23-2" > < / a > < span class = "c1" > // 在 n1 -> n2 中间插入结点 P< / span >
< a id = "__codelineno-23-3" name = "__codelineno-23-3" href = "#__codelineno-23-3" > < / a > < span class = "nx" > p< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "nx" > NewTreeNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span >
< a id = "__codelineno-23-4" name = "__codelineno-23-4" href = "#__codelineno-23-4" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > p< / span >
< a id = "__codelineno-23-5" name = "__codelineno-23-5" href = "#__codelineno-23-5" > < / a > < span class = "nx" > p< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span >
< a id = "__codelineno-23-6" name = "__codelineno-23-6" href = "#__codelineno-23-6" > < / a > < span class = "c1" > // 删除结点 P< / span >
< a id = "__codelineno-23-7" name = "__codelineno-23-7" href = "#__codelineno-23-7" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > Left< / span > < span class = "w" > < / span > < span class = "p" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-24-1" name = "__codelineno-24-1" href = "#__codelineno-24-1" > < / a > < span class = "cm" > /* 插入与删除结点 */< / span >
< a id = "__codelineno-24-2" name = "__codelineno-24-2" href = "#__codelineno-24-2" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-24-3" name = "__codelineno-24-3" href = "#__codelineno-24-3" > < / a > < span class = "c1" > // 在 n1 -> n2 中间插入结点 P< / span >
< a id = "__codelineno-24-4" name = "__codelineno-24-4" href = "#__codelineno-24-4" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-5" name = "__codelineno-24-5" href = "#__codelineno-24-5" > < / a > < span class = "nx" > P< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-24-6" name = "__codelineno-24-6" href = "#__codelineno-24-6" > < / a > < span class = "c1" > // 删除结点 P< / span >
< a id = "__codelineno-24-7" name = "__codelineno-24-7" href = "#__codelineno-24-7" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-25-1" name = "__codelineno-25-1" href = "#__codelineno-25-1" > < / a > < span class = "cm" > /* 插入与删除结点 */< / span >
< a id = "__codelineno-25-2" name = "__codelineno-25-2" href = "#__codelineno-25-2" > < / a > < span class = "kd" > const< / span > < span class = "w" > < / span > < span class = "nx" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "ow" > new< / span > < span class = "w" > < / span > < span class = "nx" > TreeNode< / span > < span class = "p" > (< / span > < span class = "mf" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-25-3" name = "__codelineno-25-3" href = "#__codelineno-25-3" > < / a > < span class = "c1" > // 在 n1 -> n2 中间插入结点 P< / span >
< a id = "__codelineno-25-4" name = "__codelineno-25-4" href = "#__codelineno-25-4" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-25-5" name = "__codelineno-25-5" href = "#__codelineno-25-5" > < / a > < span class = "nx" > P< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-25-6" name = "__codelineno-25-6" href = "#__codelineno-25-6" > < / a > < span class = "c1" > // 删除结点 P< / span >
< a id = "__codelineno-25-7" name = "__codelineno-25-7" href = "#__codelineno-25-7" > < / a > < span class = "nx" > n1< / span > < span class = "p" > .< / span > < span class = "nx" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "nx" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-26-1" name = "__codelineno-26-1" href = "#__codelineno-26-1" > < / a >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-27-1" name = "__codelineno-27-1" href = "#__codelineno-27-1" > < / a > < span class = "cm" > /* 插入与删除结点 */< / span >
< a id = "__codelineno-27-2" name = "__codelineno-27-2" href = "#__codelineno-27-2" > < / a > < span class = "n" > TreeNode< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "m" > 0< / span > < span class = "p" > );< / span >
< a id = "__codelineno-27-3" name = "__codelineno-27-3" href = "#__codelineno-27-3" > < / a > < span class = "c1" > // 在 n1 -> n2 中间插入结点 P< / span >
< a id = "__codelineno-27-4" name = "__codelineno-27-4" href = "#__codelineno-27-4" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > P< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-27-5" name = "__codelineno-27-5" href = "#__codelineno-27-5" > < / a > < span class = "n" > P< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-27-6" name = "__codelineno-27-6" href = "#__codelineno-27-6" > < / a > < span class = "c1" > // 删除结点 P< / span >
< a id = "__codelineno-27-7" name = "__codelineno-27-7" href = "#__codelineno-27-7" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > n2< / span > < span class = "p" > ;< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-28-1" name = "__codelineno-28-1" href = "#__codelineno-28-1" > < / a > < span class = "kd" > let< / span > < span class = "nv" > P< / span > < span class = "p" > =< / span > < span class = "n" > TreeNode< / span > < span class = "p" > (< / span > < span class = "n" > x< / span > < span class = "p" > :< / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span >
< a id = "__codelineno-28-2" name = "__codelineno-28-2" href = "#__codelineno-28-2" > < / a > < span class = "c1" > // 在 n1 -> n2 中间插入结点 P< / span >
< a id = "__codelineno-28-3" name = "__codelineno-28-3" href = "#__codelineno-28-3" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > P< / span >
< a id = "__codelineno-28-4" name = "__codelineno-28-4" href = "#__codelineno-28-4" > < / a > < span class = "n" > P< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > n2< / span >
< a id = "__codelineno-28-5" name = "__codelineno-28-5" href = "#__codelineno-28-5" > < / a > < span class = "c1" > // 删除结点 P< / span >
< a id = "__codelineno-28-6" name = "__codelineno-28-6" href = "#__codelineno-28-6" > < / a > < span class = "n" > n1< / span > < span class = "p" > .< / span > < span class = "kr" > left< / span > < span class = "p" > =< / span > < span class = "n" > n2< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > binary_tree.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-29-1" name = "__codelineno-29-1" href = "#__codelineno-29-1" > < / a >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< div class = "admonition note" >
< p class = "admonition-title" > Note< / p >
< p > 插入结点会改变二叉树的原有逻辑结构，删除结点往往意味着删除了该结点的所有子树。因此，二叉树中的插入与删除一般都是由一套操作配合完成的，这样才能实现有意义的操作。< / p >
< / div >
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< h2 id = "713" > 7.1.3. 常见二叉树类型< a class = "headerlink" href = "#713" title = "Permanent link" > ¶ < / a > < / h2 >
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< h3 id = "_1" > 完美二叉树< a class = "headerlink" href = "#_1" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 「完美二叉树 Perfect Binary Tree」的所有层的结点都被完全填满。在完美二叉树中， < span class = "arithmatex" > \(= h\)< / span > ，则结点总数 < span class = "arithmatex" > \(= 2^{h+1} - 1\)< / span > ，呈标准的指数级关系，反映着自然界中常见的细胞分裂。< / p >
< div class = "admonition tip" >
< p class = "admonition-title" > Tip< / p >
< p > 在中文社区中，完美二叉树常被称为「满二叉树」，请注意与完满二叉树区分。< / p >
< / div >
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< p > < img alt = "完美二叉树" src = "../binary_tree.assets/perfect_binary_tree.png" / > < / p >
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< p align = "center" > Fig. 完美二叉树 < / p >
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< h3 id = "_2" > 完全二叉树< a class = "headerlink" href = "#_2" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 「完全二叉树 Complete Binary Tree」只有最底层的结点未被填满， < / p >
< p > < strong > 完全二叉树非常适合用数组来表示< / strong > 。如果按照层序遍历序列的顺序来存储，那么空结点 < code > null< / code > 一定全部出现在序列的尾部，因此我们就可以不用存储这些 null 了。< / p >
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< p > < img alt = "完全二叉树" src = "../binary_tree.assets/complete_binary_tree.png" / > < / p >
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< p align = "center" > Fig. 完全二叉树 < / p >
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< h3 id = "_3" > 完满二叉树< a class = "headerlink" href = "#_3" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 「完满二叉树 Full Binary Tree」除了叶结点之外， < / p >
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< p > < img alt = "完满二叉树" src = "../binary_tree.assets/full_binary_tree.png" / > < / p >
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< p align = "center" > Fig. 完满二叉树 < / p >
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< h3 id = "_4" > 平衡二叉树< a class = "headerlink" href = "#_4" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 「平衡二叉树 Balanced Binary Tree」中任意结点的左子树和右子树的高度之差的绝对值 < span class = "arithmatex" > \(\leq 1\)< / span > 。< / p >
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< p > < img alt = "平衡二叉树" src = "../binary_tree.assets/balanced_binary_tree.png" / > < / p >
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< p align = "center" > Fig. 平衡二叉树 < / p >
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< h2 id = "714" > 7.1.4. 二叉树的退化< a class = "headerlink" href = "#714" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 当二叉树的每层的结点都被填满时，达到「完美二叉树」；而当所有结点都偏向一边时，二叉树退化为「链表」。< / p >
< ul >
< li > 完美二叉树是一个二叉树的“最佳状态”，可以完全发挥出二叉树“分治”的优势；< / li >
< li > 链表则是另一个极端，各项操作都变为线性操作，时间复杂度退化至 < span class = "arithmatex" > \(O(n)\)< / span > ； < / li >
< / ul >
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< p > < img alt = "二叉树的最佳与最二叉树的最佳和最差结构差情况" src = "../binary_tree.assets/binary_tree_corner_cases.png" / > < / p >
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< p align = "center" > Fig. 二叉树的最佳与最二叉树的最佳和最差结构差情况 < / p >
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< p > 如下表所示，在最佳和最差结构下，二叉树的叶结点数量、结点总数、高度等达到极大或极小值。< / p >
< div class = "center-table" >
< table >
< thead >
< tr >
< th > < / th >
< th > 完美二叉树< / th >
< th > 链表< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td > 第 < span class = "arithmatex" > \(i\)< / span > 层的结点数量< / td >
< td > < span class = "arithmatex" > \(2^{i-1}\)< / span > < / td >
< td > < span class = "arithmatex" > \(1\)< / span > < / td >
< / tr >
< tr >
< td > 树的高度为 < span class = "arithmatex" > \(h\)< / span > 时的叶结点数量< / td >
< td > < span class = "arithmatex" > \(2^h\)< / span > < / td >
< td > < span class = "arithmatex" > \(1\)< / span > < / td >
< / tr >
< tr >
< td > 树的高度为 < span class = "arithmatex" > \(h\)< / span > 时的结点总数< / td >
< td > < span class = "arithmatex" > \(2^{h+1} - 1\)< / span > < / td >
< td > < span class = "arithmatex" > \(h + 1\)< / span > < / td >
< / tr >
< tr >
< td > 树的结点总数为 < span class = "arithmatex" > \(n\)< / span > 时的高度< / td >
< td > < span class = "arithmatex" > \(\log_2 (n+1) - 1\)< / span > < / td >
< td > < span class = "arithmatex" > \(n - 1\)< / span > < / td >
< / tr >
< / tbody >
< / table >
< / div >
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< h2 id = "715" > 7.1.5. 二叉树表示方式 *< a class = "headerlink" href = "#715" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 我们一般使用二叉树的「链表表示」，即存储单位为结点 < code > TreeNode< / code > ，结点之间通过指针（引用）相连接。本文前述示例代码展示了二叉树在链表表示下的各项基本操作。< / p >
< p > 那能否可以用「数组表示」二叉树呢？答案是肯定的。先来分析一个简单案例，给定一个「完美二叉树」，将结点按照层序遍历的顺序编号（从 0 开始），那么可以推导得出父结点索引与子结点索引之间的「映射公式」：< strong > 设结点的索引为 < span class = "arithmatex" > \(i\)< / span > ，则该结点的左子结点索引为 < span class = "arithmatex" > \(2i + 1\)< / span > 、右子结点索引为 < span class = "arithmatex" > \(2i + 2\)< / span > < / strong > 。< / p >
< p > < strong > 本质上，映射公式的作用就是链表中的指针< / strong > 。对于层序遍历序列中的任意结点，我们都可以使用映射公式来访问子结点。因此，可以直接使用层序遍历序列（即数组）来表示完美二叉树。< / p >
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< p > < img alt = "完美二叉树的数组表示" src = "../binary_tree.assets/array_representation_mapping.png" / > < / p >
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< p align = "center" > Fig. 完美二叉树的数组表示 < / p >
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< p > 然而，完美二叉树只是个例，二叉树中间层往往存在许多空结点（即 < code > null< / code > ），而层序遍历序列并不包含这些空结点，并且我们无法单凭序列来猜测空结点的数量和分布位置，< strong > 即理论上存在许多种二叉树都符合该层序遍历序列< / strong > 。显然，这种情况无法使用数组来存储二叉树。< / p >
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< p > < img alt = "给定数组对应多种二叉树可能性" src = "../binary_tree.assets/array_representation_without_empty.png" / > < / p >
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< p align = "center" > Fig. 给定数组对应多种二叉树可能性 < / p >
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< p > 为了解决此问题，考虑按照完美二叉树的形式来表示所有二叉树，< strong > 即在序列中使用特殊符号来显式地表示“空位”< / strong > 。如下图所示，这样处理后，序列（数组）就可以唯一表示二叉树了。< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "4:10" > < input checked = "checked" id = "__tabbed_4_1" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_2" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_3" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_4" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_5" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_6" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_7" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_8" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_9" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_10" name = "__tabbed_4" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_4_1" > Java< / label > < label for = "__tabbed_4_2" > C++< / label > < label for = "__tabbed_4_3" > Python< / label > < label for = "__tabbed_4_4" > Go< / label > < label for = "__tabbed_4_5" > JavaScript< / label > < label for = "__tabbed_4_6" > TypeScript< / label > < label for = "__tabbed_4_7" > C< / label > < label for = "__tabbed_4_8" > C#< / label > < label for = "__tabbed_4_9" > Swift< / label > < label for = "__tabbed_4_10" > Zig< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-30-1" name = "__codelineno-30-1" href = "#__codelineno-30-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-30-2" name = "__codelineno-30-2" href = "#__codelineno-30-2" > < / a > < span class = "c1" > // 使用 int 的包装类 Integer ，就可以使用 null 来标记空位< / span >
< a id = "__codelineno-30-3" name = "__codelineno-30-3" href = "#__codelineno-30-3" > < / a > < span class = "n" > Integer< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > tree< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 15< / span > < span class = "w" > < / span > < span class = "p" > };< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-31-1" name = "__codelineno-31-1" href = "#__codelineno-31-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-31-2" name = "__codelineno-31-2" href = "#__codelineno-31-2" > < / a > < span class = "c1" > // 为了符合数据类型为 int ，使用 int 最大值标记空位< / span >
< a id = "__codelineno-31-3" name = "__codelineno-31-3" href = "#__codelineno-31-3" > < / a > < span class = "c1" > // 该方法的使用前提是没有结点的值 = INT_MAX< / span >
< a id = "__codelineno-31-4" name = "__codelineno-31-4" href = "#__codelineno-31-4" > < / a > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "n" > tree< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 15< / span > < span class = "w" > < / span > < span class = "p" > };< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-32-1" name = "__codelineno-32-1" href = "#__codelineno-32-1" > < / a > < span class = "sd" > " " " 二叉树的数组表示 " " " < / span >
< a id = "__codelineno-32-2" name = "__codelineno-32-2" href = "#__codelineno-32-2" > < / a > < span class = "c1" > # 直接使用 None 来表示空位< / span >
< a id = "__codelineno-32-3" name = "__codelineno-32-3" href = "#__codelineno-32-3" > < / a > < span class = "n" > tree< / span > < span class = "o" > =< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "kc" > None< / span > < span class = "p" > ,< / span > < span class = "mi" > 15< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
2023-03-16 22:50:42 +08:00
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-33-1" name = "__codelineno-33-1" href = "#__codelineno-33-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-33-2" name = "__codelineno-33-2" href = "#__codelineno-33-2" > < / a > < span class = "c1" > // 使用 any 类型的切片, 就可以使用 nil 来标记空位< / span >
< a id = "__codelineno-33-3" name = "__codelineno-33-3" href = "#__codelineno-33-3" > < / a > < span class = "nx" > tree< / span > < span class = "w" > < / span > < span class = "o" > :=< / span > < span class = "w" > < / span > < span class = "p" > []< / span > < span class = "kt" > any< / span > < span class = "p" > {< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 15< / span > < span class = "p" > }< / span >
2023-02-08 14:54:41 +08:00
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-34-1" name = "__codelineno-34-1" href = "#__codelineno-34-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-34-2" name = "__codelineno-34-2" href = "#__codelineno-34-2" > < / a > < span class = "c1" > // 直接使用 null 来表示空位< / span >
< a id = "__codelineno-34-3" name = "__codelineno-34-3" href = "#__codelineno-34-3" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > tree< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "mf" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 15< / span > < span class = "p" > ];< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-35-1" name = "__codelineno-35-1" href = "#__codelineno-35-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-35-2" name = "__codelineno-35-2" href = "#__codelineno-35-2" > < / a > < span class = "c1" > // 直接使用 null 来表示空位< / span >
< a id = "__codelineno-35-3" name = "__codelineno-35-3" href = "#__codelineno-35-3" > < / a > < span class = "kd" > let< / span > < span class = "w" > < / span > < span class = "nx" > tree< / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > number< / span > < span class = "w" > < / span > < span class = "o" > |< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > )[]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > [< / span > < span class = "mf" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kc" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mf" > 15< / span > < span class = "p" > ];< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-36-1" name = "__codelineno-36-1" href = "#__codelineno-36-1" > < / a >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-37-1" name = "__codelineno-37-1" href = "#__codelineno-37-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-37-2" name = "__codelineno-37-2" href = "#__codelineno-37-2" > < / a > < span class = "c1" > // 使用 int? 可空类型 ，就可以使用 null 来标记空位< / span >
< a id = "__codelineno-37-3" name = "__codelineno-37-3" href = "#__codelineno-37-3" > < / a > < span class = "kt" > int?< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > tree< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > {< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 4< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 6< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 7< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 8< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 9< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 12< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "k" > null< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 15< / span > < span class = "w" > < / span > < span class = "p" > };< / span >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-38-1" name = "__codelineno-38-1" href = "#__codelineno-38-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-38-2" name = "__codelineno-38-2" href = "#__codelineno-38-2" > < / a > < span class = "c1" > // 使用 Int? 可空类型 ，就可以使用 nil 来标记空位< / span >
< a id = "__codelineno-38-3" name = "__codelineno-38-3" href = "#__codelineno-38-3" > < / a > < span class = "kd" > let< / span > < span class = "nv" > tree< / span > < span class = "p" > :< / span > < span class = "p" > [< / span > < span class = "nb" > Int< / span > < span class = "p" > ?]< / span > < span class = "p" > =< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "mi" > 2< / span > < span class = "p" > ,< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "mi" > 4< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "mi" > 6< / span > < span class = "p" > ,< / span > < span class = "mi" > 7< / span > < span class = "p" > ,< / span > < span class = "mi" > 8< / span > < span class = "p" > ,< / span > < span class = "mi" > 9< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "mi" > 12< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "kc" > nil< / span > < span class = "p" > ,< / span > < span class = "mi" > 15< / span > < span class = "p" > ]< / span >
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< p > < img alt = "任意类型二叉树的数组表示" src = "../binary_tree.assets/array_representation_with_empty.png" / > < / p >
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< p align = "center" > Fig. 任意类型二叉树的数组表示 < / p >
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< p > 回顾「完全二叉树」的定义，其只有最底层有空结点，并且最底层的结点尽量靠左，因而所有空结点都一定出现在层序遍历序列的末尾。< strong > 因为我们先验地确定了空位的位置，所以在使用数组表示完全二叉树时，可以省略存储“空位”< / strong > 。因此，完全二叉树非常适合使用数组来表示。< / p >
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< p > < img alt = "完全二叉树的数组表示" src = "../binary_tree.assets/array_representation_complete_binary_tree.png" / > < / p >
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< p align = "center" > Fig. 完全二叉树的数组表示 < / p >
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< p > 数组表示有两个优点： 一是不需要存储指针，节省空间；二是可以随机访问结点。然而，当二叉树中的“空位”很多时，数组中只包含很少结点的数据，空间利用率很低。< / p >
< h2 id = "__comments" > 评论< / h2 >
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