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3.4. &nbsp; 字符编码 *
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3.5. &nbsp; 小结
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4.1. &nbsp; 数组
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4.2. &nbsp; 链表
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4.3. &nbsp; 列表
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4.4. &nbsp; 小结
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5.1. &nbsp; 栈
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5.2. &nbsp; 队列
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5.3. &nbsp; 双向队列
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5.4. &nbsp; 小结
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7.1. &nbsp; 二叉树
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7.2. &nbsp; 二叉树遍历
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7.3. &nbsp; 二叉树数组表示
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7.4. &nbsp; 二叉搜索树
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7.5. &nbsp; AVL 树 *
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8.1. &nbsp; 堆
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8.2. &nbsp; 建堆操作
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8.3. &nbsp; Top-K 问题（New）
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8.4. &nbsp; 小结
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9.1. &nbsp; 图
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9.2. &nbsp; 图基础操作
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9.3. &nbsp; 图的遍历
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10.1. &nbsp; 二分查找
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10.2. &nbsp; 二分查找边界
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10.3. &nbsp; 哈希优化策略
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10.4. &nbsp; 重识搜索算法
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11.1. &nbsp; 排序算法
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11.2. &nbsp; 选择排序
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11.3. &nbsp; 冒泡排序
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11.4. &nbsp; 插入排序
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11.5. &nbsp; 快速排序
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11.6. &nbsp; 归并排序
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11.7. &nbsp; 堆排序
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11.8. &nbsp; 桶排序
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11.9. &nbsp; 计数排序
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11.10. &nbsp; 基数排序
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12.3. &nbsp; 构建树问题（New）
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13.1. &nbsp; 回溯算法
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13.2. &nbsp; 全排列问题
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13.3. &nbsp; 子集和问题
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13.4. &nbsp; N 皇后问题
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14.4. &nbsp; 0-1 背包问题（New）
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<h1 id="76">7.6. &nbsp; 小结<a class="headerlink" href="#76" title="Permanent link">&para;</a></h1>
<ul>
<li>二叉树是一种非线性数据结构，体现“一分为二”的分治逻辑。每个二叉树节点包含一个值以及两个指针，分别指向其左子节点和右子节点。</li>
<li>对于二叉树中的某个节点，其左（右）子节点及其以下形成的树被称为该节点的左（右）子树。</li>
<li>二叉树的相关术语包括根节点、叶节点、层、度、边、高度和深度等。</li>
<li>二叉树的初始化、节点插入和节点删除操作与链表操作方法类似。</li>
<li>常见的二叉树类型有完美二叉树、完全二叉树、满二叉树和平衡二叉树。完美二叉树是最理想的状态，而链表是退化后的最差状态。</li>
<li>二叉树可以用数组表示，方法是将节点值和空位按层序遍历顺序排列，并根据父节点与子节点之间的索引映射关系来实现指针。</li>
<li>二叉树的层序遍历是一种广度优先搜索方法，它体现了“一圈一圈向外”的分层遍历方式，通常通过队列来实现。</li>
<li>前序、中序、后序遍历皆属于深度优先搜索，它们体现了“走到尽头，再回头继续”的回溯遍历方式，通常使用递归来实现。</li>
<li>二叉搜索树是一种高效的元素查找数据结构，其查找、插入和删除操作的时间复杂度均为 <span class="arithmatex">\(O(\log n)\)</span> 。当二叉搜索树退化为链表时，各项时间复杂度会劣化至 <span class="arithmatex">\(O(n)\)</span> 。</li>
<li>AVL 树，也称为平衡二叉搜索树，它通过旋转操作，确保在不断插入和删除节点后，树仍然保持平衡。</li>
<li>AVL 树的旋转操作包括右旋、左旋、先右旋再左旋、先左旋再右旋。在插入或删除节点后，AVL 树会从底向顶执行旋转操作，使树重新恢复平衡。</li>
</ul>
<h2 id="761-q-a">7.6.1. &nbsp; Q &amp; A<a class="headerlink" href="#761-q-a" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">对于只有一个节点的二叉树，树的高度和根节点的深度都是 <span class="arithmatex">\(0\)</span> 吗？</p>
<p>是的，因为高度和深度通常定义为“走过边的数量”。</p>
</div>
<div class="admonition question">
<p class="admonition-title">二叉树中的插入与删除一般都是由一套操作配合完成的，这里的“一套操作”指什么呢？可以理解为资源的子节点的资源释放吗？</p>
<p>拿二叉搜索树来举例，删除节点操作要分为三种情况处理，其中每种情况都需要进行多个步骤的节点操作。</p>
</div>
<div class="admonition question">
<p class="admonition-title">为什么 DFS 遍历二叉树有前、中、后三种顺序，分别有什么用呢？</p>
<p>DFS 的前、中、后序遍历和访问数组的顺序类似，是遍历二叉树的基本方法，利用这三种遍历方法，我们可以得到一个特定顺序的遍历结果。例如在二叉搜索树中，由于结点大小满足 <code>左子结点值 &lt; 根结点值 &lt; 右子结点值</code> ，因此我们只要按照 <code>左-&gt;根-&gt;右</code> 的优先级遍历树，就可以获得有序的节点序列。</p>
</div>
<div class="admonition question">
<p class="admonition-title">右旋操作是处理失衡节点 <code>node</code> , <code>child</code> , <code>grand_child</code> 之间的关系，那 <code>node</code> 的父节点和 <code>node</code> 原来的连接不需要维护吗？右旋操作后岂不是断掉了？</p>
<p>我们需要从递归的视角来看这个问题。右旋操作 <code>right_rotate(root)</code> 传入的是子树的根节点，最终 <code>return child</code> 返回旋转之后的子树的根节点。子树的根节点和其父节点的连接是在该函数返回后完成的，不属于右旋操作的维护范围。</p>
</div>
<div class="admonition question">
<p class="admonition-title">在 C++ 中，函数被划分到 <code>private</code> 和 <code>public</code> 中，这方面有什么考量吗？为什么要将 <code>height()</code> 函数和 <code>updateHeight()</code> 函数分别放在 <code>public</code> 和 <code>private</code> 中呢？</p>
<p>主要看方法的使用范围，如果方法只在类内部使用，那么就设计为 <code>private</code> 。例如，用户单独调用 <code>updateHeight()</code> 是没有意义的，它只是插入、删除操作中的一步。而 <code>height()</code> 是访问结点高度，类似于 <code>vector.size()</code> ，因此设置成 <code>public</code> 以便使用。</p>
</div>
<div class="admonition question">
<p class="admonition-title">请问如何从一组输入数据构建一个二叉搜索树？根节点的选择是不是很重要？</p>
<p>是的，构建树的方法是 <code>build_tree()</code> ，已在源代码中给出。至于根节点的选择，我们通常会将输入数据排序，然后用中点元素作为根节点，再递归地构建左右子树。这样做可以最大程度保证树的平衡性。</p>
</div>
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