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7.3. 二叉树数组表示
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0. 前言
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< a href = "../../chapter_preface/about_the_book/" class = "md-nav__link" >
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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. 初识算法
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1. 初识算法
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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
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2. 复杂度
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2.1. 算法效率评估
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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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2.4. 小结
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< a href = "../../chapter_data_structure/" class = "md-nav__link " >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M12 3C7.58 3 4 4.79 4 7v10c0 2.21 3.59 4 8 4s8-1.79 8-4V7c0-2.21-3.58-4-8-4m6 14c0 .5-2.13 2-6 2s-6-1.5-6-2v-2.23c1.61.78 3.72 1.23 6 1.23s4.39-.45 6-1.23V17m0-4.55c-1.3.95-3.58 1.55-6 1.55s-4.7-.6-6-1.55V9.64c1.47.83 3.61 1.36 6 1.36s4.53-.53 6-1.36v2.81M12 9C8.13 9 6 7.5 6 7s2.13-2 6-2 6 1.5 6 2-2.13 2-6 2Z" / > < / svg >
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3. 数据结构
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3. 数据结构
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3.1. 数据结构分类
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3.2. 基本数据类型
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3.3. 数字编码 *
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3.4. 字符编码 *
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3.5. 小结
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4. 数组与链表
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4. 数组与链表
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4.1. 数组
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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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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_stack_and_queue/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z" / > < / svg >
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5. 栈与队列
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5. 栈与队列
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/stack/" class = "md-nav__link" >
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5.1. 栈
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/queue/" class = "md-nav__link" >
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5.2. 队列
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/deque/" class = "md-nav__link" >
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5.3. 双向队列
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
5.4. 小结
< / span >
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< / a >
< / li >
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< / ul >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_hashing/" class = "md-nav__link " >
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< span class = "md-ellipsis" >
6. 散列表
< / span >
< / a >
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< / div >
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< label class = "md-nav__title" for = "__nav_7" >
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6. 散列表
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_map/" class = "md-nav__link" >
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6.1. 哈希表
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_collision/" class = "md-nav__link" >
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6.2. 哈希冲突
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_algorithm/" class = "md-nav__link" >
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6.3. 哈希算法
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
6.4. 小结
< / span >
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< / a >
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< / ul >
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< li class = "md-nav__item md-nav__item--active md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_8" checked >
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< a href = "../" class = "md-nav__link " >
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< span class = "md-ellipsis" >
7. 树
< / span >
< / a >
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< / label >
< / div >
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< span class = "md-nav__icon md-icon" > < / span >
7. 树
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../binary_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.1. 二叉树
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../binary_tree_traversal/" class = "md-nav__link" >
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7.2. 二叉树遍历
< / span >
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< / a >
< / li >
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< li class = "md-nav__item md-nav__item--active" >
< input class = "md-nav__toggle md-toggle" type = "checkbox" id = "__toc" >
< label class = "md-nav__link md-nav__link--active" for = "__toc" >
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< span class = "md-ellipsis" >
7.3. 二叉树数组表示
< / span >
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< / label >
< a href = "./" class = "md-nav__link md-nav__link--active" >
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< span class = "md-ellipsis" >
7.3. 二叉树数组表示
< / span >
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< / a >
< 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" >
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< a href = "#731" class = "md-nav__link" >
7.3.1. 表示完美二叉树
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< / a >
< / li >
< li class = "md-nav__item" >
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< a href = "#732" class = "md-nav__link" >
7.3.2. 表示任意二叉树
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< / a >
< / li >
< li class = "md-nav__item" >
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< a href = "#733" class = "md-nav__link" >
7.3.3. 优势与局限性
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< / ul >
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< / li >
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< li class = "md-nav__item" >
< a href = "../binary_search_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.4. 二叉搜索树
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../avl_tree/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.5. AVL 树 *
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
7.6. 小结
< / span >
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< / li >
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< / ul >
< / nav >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_9" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_heap/" class = "md-nav__link " >
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8. 堆
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< / label >
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< label class = "md-nav__title" for = "__nav_9" >
< span class = "md-nav__icon md-icon" > < / span >
8. 堆
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< a href = "../../chapter_heap/heap/" class = "md-nav__link" >
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8.1. 堆
< / span >
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< / a >
< / li >
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< a href = "../../chapter_heap/build_heap/" class = "md-nav__link" >
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8.2. 建堆操作
< / span >
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< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_heap/top_k/" class = "md-nav__link" >
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8.3. Top-K 问题
< / span >
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< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_heap/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
8.4. 小结
< / span >
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< / li >
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< / ul >
< / nav >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_10" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_graph/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z" / > < / svg >
< span class = "md-ellipsis" >
9. 图
< / span >
< / a >
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< label class = "md-nav__link " for = "__nav_10" >
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_10_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_10" >
< span class = "md-nav__icon md-icon" > < / span >
9. 图
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.1. 图
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.2. 图基础操作
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph_traversal/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.3. 图的遍历
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_graph/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
9.4. 小结
< / span >
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< / a >
< / li >
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< / ul >
< / nav >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_11" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_searching/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z" / > < / svg >
< span class = "md-ellipsis" >
10. 搜索
< / span >
< / a >
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< label class = "md-nav__link " for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_11_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
10. 搜索
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.1. 二分查找
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/binary_search_edge/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.2. 二分查找边界
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/replace_linear_by_hashing/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.3. 哈希优化策略
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/searching_algorithm_revisited/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.4. 重识搜索算法
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
10.5. 小结
< / span >
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< / a >
< / li >
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< / ul >
< / nav >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_12" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_sorting/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z" / > < / svg >
< span class = "md-ellipsis" >
11. 排序
< / span >
< / a >
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< label class = "md-nav__link " for = "__nav_12" >
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_12_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_12" >
< span class = "md-nav__icon md-icon" > < / span >
11. 排序
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/sorting_algorithm/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.1. 排序算法
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/selection_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.2. 选择排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.3. 冒泡排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/insertion_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.4. 插入排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/quick_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.5. 快速排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/merge_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.6. 归并排序
< / span >
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< / a >
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< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/heap_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.7. 堆排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/bucket_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.8. 桶排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/counting_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.9. 计数排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/radix_sort/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.10. 基数排序
< / span >
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< / a >
< / li >
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< li class = "md-nav__item" >
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< a href = "../../chapter_sorting/summary/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
11.11. 小结
< / span >
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< / a >
< / li >
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< / ul >
< / nav >
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< / li >
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< li class = "md-nav__item md-nav__item--nested" >
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_13" >
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_divide_and_conquer/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M12 13H7v5h5v2H5V10h2v1h5v2M8 4v2H4V4h4m2-2H2v6h8V2m10 9v2h-4v-2h4m2-2h-8v6h8V9m-2 9v2h-4v-2h4m2-2h-8v6h8v-6Z" / > < / svg >
< span class = "md-ellipsis" >
12. 分治
< / span >
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< span class = "md-status md-status--new" title = "最近添加" >
< / span >
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< / a >
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< label class = "md-nav__link " for = "__nav_13" >
< span class = "md-nav__icon md-icon" > < / span >
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14.1. 初探动态规划
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7.3.1. 表示完美二叉树
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< / a >
< / li >
< li class = "md-nav__item" >
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< a href = "#732" class = "md-nav__link" >
7.3.2. 表示任意二叉树
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< / a >
< / li >
< li class = "md-nav__item" >
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< a href = "#733" class = "md-nav__link" >
7.3.3. 优势与局限性
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< / a >
< / li >
< / ul >
< / nav >
< / div >
< / div >
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< 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/array_representation_of_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 = "73" > 7.3. 二叉树数组表示< a class = "headerlink" href = "#73" title = "Permanent link" > ¶ < / a > < / h1 >
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< p > 在链表表示下，二叉树的存储单元为节点 < code > TreeNode< / code > ，节点之间通过指针相连接。在上节中，我们学习了在链表表示下的二叉树的各项基本操作。< / p >
< p > 那么，能否用「数组」来表示二叉树呢？答案是肯定的。< / p >
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< h2 id = "731" > 7.3.1. 表示完美二叉树< a class = "headerlink" href = "#731" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 先分析一个简单案例，给定一个完美二叉树，我们将节点按照层序遍历的顺序编号（从 < span class = "arithmatex" > \(0\)< / span > 开始），此时每个节点都对应唯一的索引。< / p >
< p > 根据层序遍历的特性，我们可以推导出父节点索引与子节点索引之间的“映射公式”：< strong > 若节点的索引为 < span class = "arithmatex" > \(i\)< / span > ，则该节点的左子节点索引为 < span class = "arithmatex" > \(2i + 1\)< / span > ，右子节点索引为 < span class = "arithmatex" > \(2i + 2\)< / span > < / strong > 。< / p >
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< p > < img alt = "完美二叉树的数组表示" src = "../array_representation_of_tree.assets/array_representation_binary_tree.png" / > < / p >
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< p align = "center" > Fig. 完美二叉树的数组表示 < / p >
< p > < strong > 映射公式的作用相当于链表中的指针< / strong > 。如果我们将节点按照层序遍历的顺序存储在一个数组中，那么对于数组中的任意节点，我们都可以通过映射公式来访问其子节点。< / p >
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< h2 id = "732" > 7.3.2. 表示任意二叉树< a class = "headerlink" href = "#732" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 然而，完美二叉树只是一个特例。在二叉树的中间层，通常存在许多 < span class = "arithmatex" > \(\text{None}\)< / span > ，而层序遍历序列并不包含这些 < span class = "arithmatex" > \(\text{None}\)< / span > 。我们无法仅凭该序列来推测 < span class = "arithmatex" > \(\text{None}\)< / span > 的数量和分布位置，< strong > 这意味着存在多种二叉树结构都符合该层序遍历序列< / strong > 。显然在这种情况下，上述的数组表示方法已经失效。< / p >
< p > < img alt = "层序遍历序列对应多种二叉树可能性" src = "../array_representation_of_tree.assets/array_representation_without_empty.png" / > < / p >
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< p align = "center" > Fig. 层序遍历序列对应多种二叉树可能性 < / p >
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< p > 为了解决此问题，< strong > 我们可以考虑在层序遍历序列中显式地写出所有 < span class = "arithmatex" > \(\text{None}\)< / span > < / strong > 。如下图所示，这样处理后，层序遍历序列就可以唯一表示二叉树了。< / p >
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< div class = "tabbed-set tabbed-alternate" data-tabs = "1:11" > < 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" / > < input id = "__tabbed_1_11" 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 > < label for = "__tabbed_1_11" > Dart< / label > < / div >
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< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / 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-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / 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-1-1" name = "__codelineno-1-1" href = "#__codelineno-1-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
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< a id = "__codelineno-1-2" name = "__codelineno-1-2" href = "#__codelineno-1-2" > < / a > < span class = "c1" > // 使用 int 最大值标记空位，因此要求节点值不能为 INT_MAX< / span >
< a id = "__codelineno-1-3" name = "__codelineno-1-3" href = "#__codelineno-1-3" > < / 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 = "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 = "p" > };< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "c1" > # 二叉树的数组表示< / span >
< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a > < span class = "c1" > # 直接使用 None 来表示空位< / span >
< a id = "__codelineno-2-3" name = "__codelineno-2-3" href = "#__codelineno-2-3" > < / 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" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-3-1" name = "__codelineno-3-1" href = "#__codelineno-3-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-3-2" name = "__codelineno-3-2" href = "#__codelineno-3-2" > < / 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-3-3" name = "__codelineno-3-3" href = "#__codelineno-3-3" > < / 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 >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-4-1" name = "__codelineno-4-1" href = "#__codelineno-4-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-4-2" name = "__codelineno-4-2" href = "#__codelineno-4-2" > < / a > < span class = "c1" > // 直接使用 null 来表示空位< / span >
< a id = "__codelineno-4-3" name = "__codelineno-4-3" href = "#__codelineno-4-3" > < / 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-5-1" name = "__codelineno-5-1" href = "#__codelineno-5-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-5-2" name = "__codelineno-5-2" href = "#__codelineno-5-2" > < / a > < span class = "c1" > // 直接使用 null 来表示空位< / span >
< a id = "__codelineno-5-3" name = "__codelineno-5-3" href = "#__codelineno-5-3" > < / 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" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-6-1" name = "__codelineno-6-1" href = "#__codelineno-6-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-6-2" name = "__codelineno-6-2" href = "#__codelineno-6-2" > < / a > < span class = "c1" > // 使用 int 最大值标记空位，因此要求节点值不能为 INT_MAX< / span >
< a id = "__codelineno-6-3" name = "__codelineno-6-3" href = "#__codelineno-6-3" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > tree< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / 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 = "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 = "p" > };< / span >
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< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-7-1" name = "__codelineno-7-1" href = "#__codelineno-7-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-7-2" name = "__codelineno-7-2" href = "#__codelineno-7-2" > < / 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-7-3" name = "__codelineno-7-3" href = "#__codelineno-7-3" > < / 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 >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-8-1" name = "__codelineno-8-1" href = "#__codelineno-8-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-8-2" name = "__codelineno-8-2" href = "#__codelineno-8-2" > < / 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-8-3" name = "__codelineno-8-3" href = "#__codelineno-8-3" > < / 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 >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
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< / code > < / pre > < / div >
< / div >
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< div class = "tabbed-block" >
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< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-10-1" name = "__codelineno-10-1" href = "#__codelineno-10-1" > < / a > < span class = "cm" > /* 二叉树的数组表示 */< / span >
< a id = "__codelineno-10-2" name = "__codelineno-10-2" href = "#__codelineno-10-2" > < / 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-10-3" name = "__codelineno-10-3" href = "#__codelineno-10-3" > < / a > < span class = "n" > List< / 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 = "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 = "kc" > 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 = "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 = "m" > 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 = "m" > 15< / span > < span class = "p" > ];< / span >
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< / code > < / pre > < / div >
< / div >
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< / div >
< / div >
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< p > < img alt = "任意类型二叉树的数组表示" src = "../array_representation_of_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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< h2 id = "733" > 7.3.3. 优势与局限性< a class = "headerlink" href = "#733" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 二叉树的数组表示存在以下优点：< / p >
< ul >
< li > 数组存储在连续的内存空间中，缓存友好，访问与遍历速度较快；< / li >
< li > 不需要存储指针，比较节省空间；< / li >
< li > 允许随机访问节点；< / li >
< / ul >
< p > 然而，数组表示也具有一些局限性：< / p >
< ul >
< li > 数组存储需要连续内存空间，因此不适合存储数据量过大的树。< / li >
< li > 增删节点需要通过数组插入与删除操作实现，效率较低；< / li >
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< li > 当二叉树中存在大量 < span class = "arithmatex" > \(\text{None}\)< / span > 时，数组中包含的节点数据比重较低，空间利用率较低。< / li >
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< / ul >
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< p > < strong > 完全二叉树非常适合使用数组来表示< / strong > 。回顾完全二叉树的定义，< span class = "arithmatex" > \(\text{None}\)< / span > 只出现在最底层且靠右的位置，< strong > 这意味着所有 < span class = "arithmatex" > \(\text{None}\)< / span > 一定出现在层序遍历序列的末尾< / strong > 。因此，在使用数组表示完全二叉树时，可以省略存储所有 < span class = "arithmatex" > \(\text{None}\)< / span > 。< / p >
< p > < img alt = "完全二叉树的数组表示" src = "../array_representation_of_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 >
< h2 id = "__comments" > 评论< / h2 >
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