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<title>7.4.   AVL 树 * - Hello 算法</title>
<title>8.4.   AVL 树 * - Hello 算法</title>
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<div class="md-header__topic" data-md-component="header-topic">
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7.4. &nbsp; AVL 树 *
8.4. &nbsp; AVL 树 *
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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<li class="md-nav__item">
<a href="../../chapter_computational_complexity/space_time_tradeoff/" class="md-nav__link">
2.4. &nbsp; 权衡时间与空间
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
2.5. &nbsp; 小结
2.4. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 二分查找
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6. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 二分查找
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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7. &nbsp; &nbsp; 散列表
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7. &nbsp; &nbsp; 散列表
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. &nbsp; 哈希表
7.1. &nbsp; 哈希表
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
6.2. &nbsp; 哈希冲突处理
7.2. &nbsp; 哈希冲突处理
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
6.3. &nbsp; 小结
7.3. &nbsp; 小结
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 树
<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 树
<span class="md-nav__icon md-icon"></span>
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7. &nbsp; &nbsp; 树
8. &nbsp; &nbsp; 树
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<li class="md-nav__item">
<a href="../binary_tree/" class="md-nav__link">
7.1. &nbsp; 二叉树
8.1. &nbsp; 二叉树
</a>
</li>
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<li class="md-nav__item">
<a href="../binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
8.2. &nbsp; 二叉树遍历
</a>
</li>
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<li class="md-nav__item">
<a href="../binary_search_tree/" class="md-nav__link">
7.3. &nbsp; 二叉搜索树
8.3. &nbsp; 二叉搜索树
</a>
</li>
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<label class="md-nav__link md-nav__link--active" for="__toc">
7.4. &nbsp; AVL 树 *
8.4. &nbsp; AVL 树 *
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
7.4. &nbsp; AVL 树 *
8.4. &nbsp; AVL 树 *
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
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7.4.1. &nbsp; AVL 树常见术语
<a href="#841-avl" class="md-nav__link">
8.4.1. &nbsp; AVL 树常见术语
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7.4.2. &nbsp; AVL 树旋转
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8.4.2. &nbsp; AVL 树旋转
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<nav class="md-nav" aria-label="7.4.2. &nbsp; AVL 树旋转">
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</li>
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7.4.3. &nbsp; AVL 树常用操作
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8.4.3. &nbsp; AVL 树常用操作
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<nav class="md-nav" aria-label="7.4.3. &nbsp; AVL 树常用操作">
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</li>
<li class="md-nav__item">
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7.4.4. &nbsp; AVL 树典型应用
<a href="#844-avl" class="md-nav__link">
8.4.4. &nbsp; AVL 树典型应用
</a>
</li>
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<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
7.5. &nbsp; 小结
</a>
</li>
</ul>
</nav>
</li>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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8. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
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8. &nbsp; &nbsp; 堆
</label>
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<li class="md-nav__item">
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8.1. &nbsp; 堆
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
8.2. &nbsp; 建堆操作 *
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
8.3. &nbsp; 小结
8.5. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. &nbsp; &nbsp; 图
9. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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9. &nbsp; &nbsp; 图
9. &nbsp; &nbsp; 堆
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<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
9.1. &nbsp; 图
<a href="../../chapter_heap/heap/" class="md-nav__link">
9.1. &nbsp; 堆
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
9.2. &nbsp; 图基础操作
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9.2. &nbsp; 建堆操作 *
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
9.3. &nbsp; 图的遍历
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</li>
<li class="md-nav__item">
<a href="../../chapter_graph/summary/" class="md-nav__link">
9.4. &nbsp; 小结
<a href="../../chapter_heap/summary/" class="md-nav__link">
9.3. &nbsp; 小结
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. &nbsp; &nbsp; 查找算法
10. &nbsp; &nbsp; 图
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10. &nbsp; &nbsp; 图
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<li class="md-nav__item">
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
10.1. &nbsp; 线性查找
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10.1. &nbsp; 图
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.2. &nbsp; 二分查找
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10.2. &nbsp; 图基础操作
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<li class="md-nav__item">
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
10.3. &nbsp; 哈希查找
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10.3. &nbsp; 图的遍历
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10.4. &nbsp; 小结
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<li class="md-nav__item">
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
11.6. &nbsp; 桶排序（New）
11.6. &nbsp; 桶排序
</a>
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<li class="md-nav__item">
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11.7. &nbsp; 计数排序（New）
11.7. &nbsp; 计数排序
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11.8. &nbsp; 基数排序
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
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12. &nbsp; &nbsp; 回溯算法
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<li class="md-nav__item">
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12.1. &nbsp; 回溯算法（New）
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
12.1. &nbsp; 搜索算法（New）
</a>
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12.2. &nbsp; 哈希优化策略
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12.3. &nbsp; 小结
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13. &nbsp; &nbsp; 附录
13. &nbsp; &nbsp; 回溯算法
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13.1. &nbsp; 编程环境安装
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<a href="../../chapter_appendix/contribution/" class="md-nav__link">
13.2. &nbsp; 一起参与创作
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13.1. &nbsp; 回溯算法（New）
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14.1. &nbsp; 编程环境安装
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14.2. &nbsp; 一起参与创作
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参考文献
</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#741-avl" class="md-nav__link">
7.4.1. &nbsp; AVL 树常见术语
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8.4.1. &nbsp; AVL 树常见术语
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</li>
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</li>
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7.4.4. &nbsp; AVL 树典型应用
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<h1 id="74-avl">7.4. &nbsp; AVL 树 *<a class="headerlink" href="#74-avl" title="Permanent link">&para;</a></h1>
<h1 id="84-avl">8.4. &nbsp; AVL 树 *<a class="headerlink" href="#84-avl" title="Permanent link">&para;</a></h1>
<p>在二叉搜索树章节中，我们提到了在多次插入和删除操作后，二叉搜索树可能退化为链表。这种情况下，所有操作的时间复杂度将从 <span class="arithmatex">\(O(\log n)\)</span> 恶化为 <span class="arithmatex">\(O(n)\)</span>。</p>
<p>如下图所示，经过两次删除节点操作，这个二叉搜索树便会退化为链表。</p>
<p><img alt="AVL 树在删除节点后发生退化" src="../avl_tree.assets/avltree_degradation_from_removing_node.png" /></p>
@@ -2009,7 +2032,7 @@
<p align="center"> Fig. AVL 树在插入节点后发生退化 </p>
<p>G. M. Adelson-Velsky 和 E. M. Landis 在其 1962 年发表的论文 "An algorithm for the organization of information" 中提出了「AVL 树」。论文中详细描述了一系列操作，确保在持续添加和删除节点后，AVL 树不会退化，从而使得各种操作的时间复杂度保持在 <span class="arithmatex">\(O(\log n)\)</span> 级别。换句话说，在需要频繁进行增删查改操作的场景中，AVL 树能始终保持高效的数据操作性能，具有很好的应用价值。</p>
<h2 id="741-avl">7.4.1. &nbsp; AVL 树常见术语<a class="headerlink" href="#741-avl" title="Permanent link">&para;</a></h2>
<h2 id="841-avl">8.4.1. &nbsp; AVL 树常见术语<a class="headerlink" href="#841-avl" title="Permanent link">&para;</a></h2>
<p>「AVL 树」既是二叉搜索树也是平衡二叉树，同时满足这两类二叉树的所有性质，因此也被称为「平衡二叉搜索树」。</p>
<h3 id="_1">节点高度<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<p>在操作 AVL 树时，我们需要获取节点的高度，因此需要为 AVL 树的节点类添加 <code>height</code> 变量。</p>
@@ -2382,7 +2405,7 @@
<p class="admonition-title">Note</p>
<p>设平衡因子为 <span class="arithmatex">\(f\)</span> ，则一棵 AVL 树的任意节点的平衡因子皆满足 <span class="arithmatex">\(-1 \le f \le 1\)</span> 。</p>
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<h2 id="742-avl">7.4.2. &nbsp; AVL 树旋转<a class="headerlink" href="#742-avl" title="Permanent link">&para;</a></h2>
<h2 id="842-avl">8.4.2. &nbsp; AVL 树旋转<a class="headerlink" href="#842-avl" title="Permanent link">&para;</a></h2>
<p>AVL 树的特点在于「旋转 Rotation」操作，它能够在不影响二叉树的中序遍历序列的前提下，使失衡节点重新恢复平衡。换句话说，<strong>旋转操作既能保持树的「二叉搜索树」属性，也能使树重新变为「平衡二叉树」</strong>。</p>
<p>我们将平衡因子绝对值 <span class="arithmatex">\(&gt; 1\)</span> 的节点称为「失衡节点」。根据节点失衡情况的不同，旋转操作分为四种：右旋、左旋、先右旋后左旋、先左旋后右旋。下面我们将详细介绍这些旋转操作。</p>
<h3 id="_3">右旋<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h3>
@@ -3073,7 +3096,7 @@
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<h2 id="743-avl">7.4.3. &nbsp; AVL 树常用操作<a class="headerlink" href="#743-avl" title="Permanent link">&para;</a></h2>
<h2 id="843-avl">8.4.3. &nbsp; AVL 树常用操作<a class="headerlink" href="#843-avl" title="Permanent link">&para;</a></h2>
<h3 id="_8">插入节点<a class="headerlink" href="#_8" title="Permanent link">&para;</a></h3>
<p>「AVL 树」的节点插入操作与「二叉搜索树」在主体上类似。唯一的区别在于，在 AVL 树中插入节点后，从该节点到根节点的路径上可能会出现一系列失衡节点。因此，<strong>我们需要从这个节点开始，自底向上执行旋转操作，使所有失衡节点恢复平衡</strong>。</p>
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@@ -3723,7 +3746,7 @@
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<h3 id="_10">查找节点<a class="headerlink" href="#_10" title="Permanent link">&para;</a></h3>
<p>AVL 树的节点查找操作与二叉搜索树一致，在此不再赘述。</p>
<h2 id="744-avl">7.4.4. &nbsp; AVL 树典型应用<a class="headerlink" href="#744-avl" title="Permanent link">&para;</a></h2>
<h2 id="844-avl">8.4.4. &nbsp; AVL 树典型应用<a class="headerlink" href="#844-avl" title="Permanent link">&para;</a></h2>
<ul>
<li>组织和存储大型数据，适用于高频查找、低频增删的场景；</li>
<li>用于构建数据库中的索引系统；</li>
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