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chapter_tree/array_representation_of_tree/index.html
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<a href="../../chapter_hello_algo/" class="md-nav__link ">
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<span class="md-ellipsis">
序
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</a>
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序
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</nav>
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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</a>
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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</a>
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第 8 章 &nbsp; 堆
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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</a>
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第 9 章 &nbsp; 图
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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</a>
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第 10 章 &nbsp; 搜索
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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</a>
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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<a href="../../chapter_paperbook/" class="md-nav__link ">
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纸质书
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纸质书
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<div class="md-copyright">
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Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
</div>
chapter_tree/avl_tree/index.html
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序
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 10 章 &nbsp; 搜索
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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<a href="../../chapter_paperbook/" class="md-nav__link ">
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纸质书
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<p><a class="glightbox" href="../avl_tree.assets/avltree_degradation_from_inserting_node.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="AVL 树在插入节点后发生退化" class="animation-figure" src="../avl_tree.assets/avltree_degradation_from_inserting_node.png" /></a></p>
<p align="center"> 图 7-25 &nbsp; AVL 树在插入节点后发生退化 </p>
<p>1962 年 G. M. Adelson-Velsky 和 E. M. Landis 在论文“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>
<p>1962 年 G. M. Adelson-Velsky 和 E. M. Landis 在论文“An algorithm for the organization of information”中提出了<u>AVL（树）</u>。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="751-avl">7.5.1 &nbsp; AVL 树常见术语<a class="headerlink" href="#751-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，因此是一种「平衡二叉搜索树 balanced binary search tree」。</p>
<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，因此是一种<u>平衡二叉搜索树（balanced binary search tree）</u>。</p>
<h3 id="1">1. &nbsp; 节点高度<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p>由于 AVL 树的相关操作需要获取节点高度，因此我们需要为节点类添加 <code>height</code> 变量：</p>
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<h3 id="2">2. &nbsp; 节点平衡因子<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p>节点的「平衡因子 balance factor」定义为节点左子树的高度减去右子树的高度，同时规定空节点的平衡因子为 <span class="arithmatex">\(0\)</span> 。我们同样将获取节点平衡因子的功能封装成函数，方便后续使用：</p>
<p>节点的<u>平衡因子（balance factor）</u>定义为节点左子树的高度减去右子树的高度，同时规定空节点的平衡因子为 <span class="arithmatex">\(0\)</span> 。我们同样将获取节点平衡因子的功能封装成函数，方便后续使用：</p>
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 16 章 &nbsp; 附录
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参考文献
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<a href="../../chapter_paperbook/" class="md-nav__link ">
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纸质书
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纸质书
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<!-- Page content -->
<h1 id="74">7.4 &nbsp; 二叉搜索树<a class="headerlink" href="#74" title="Permanent link">&para;</a></h1>
<p>如图 7-16 所示，「二叉搜索树 binary search tree」满足以下条件。</p>
<p>如图 7-16 所示，<u>二叉搜索树（binary search tree）</u>满足以下条件。</p>
<ol>
<li>对于根节点，左子树中所有节点的值 <span class="arithmatex">\(&lt;\)</span> 根节点的值 <span class="arithmatex">\(&lt;\)</span> 右子树中所有节点的值。</li>
<li>任意节点的左、右子树也是二叉搜索树，即同样满足条件 <code>1.</code> 。</li>
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<div class="md-copyright">
<div class="md-copyright__highlight">
Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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chapter_tree/binary_tree/index.html
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<span class="md-ellipsis">
序
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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</a>
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第 8 章 &nbsp; 堆
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<label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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第 9 章 &nbsp; 图
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第 10 章 &nbsp; 搜索
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第 11 章 &nbsp; 排序
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</a>
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第 12 章 &nbsp; 分治
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@@ -2806,7 +2863,7 @@
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</a>
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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<h1 id="71">7.1 &nbsp; 二叉树<a class="headerlink" href="#71" title="Permanent link">&para;</a></h1>
<p>「二叉树 binary tree」是一种非线性数据结构，代表“祖先”与“后代”之间的派生关系，体现了“一分为二”的分治逻辑。与链表类似，二叉树的基本单元是节点，每个节点包含值、左子节点引用和右子节点引用。</p>
<p><u>二叉树（binary tree）</u>是一种非线性数据结构，代表“祖先”与“后代”之间的派生关系，体现了“一分为二”的分治逻辑。与链表类似，二叉树的基本单元是节点，每个节点包含值、左子节点引用和右子节点引用。</p>
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<p>每个节点都有两个引用（指针），分别指向「左子节点 left-child node」和「右子节点 right-child node」，该节点被称为这两个子节点的「父节点 parent node」。当给定一个二叉树的节点时，我们将该节点的左子节点及其以下节点形成的树称为该节点的「左子树 left subtree」，同理可得「右子树 right subtree」。</p>
<p>每个节点都有两个引用（指针），分别指向<u>左子节点（left-child node）</u>和<u>右子节点（right-child node）</u>，该节点被称为这两个子节点的<u>父节点（parent node）</u>。当给定一个二叉树的节点时，我们将该节点的左子节点及其以下节点形成的树称为该节点的<u>左子树（left subtree）</u>，同理可得<u>右子树（right subtree）</u>。</p>
<p><strong>在二叉树中，除叶节点外，其他所有节点都包含子节点和非空子树</strong>。如图 7-1 所示，如果将“节点 2”视为父节点，则其左子节点和右子节点分别是“节点 4”和“节点 5”，左子树是“节点 4 及其以下节点形成的树”，右子树是“节点 5 及其以下节点形成的树”。</p>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="父节点、子节点、子树" class="animation-figure" src="../binary_tree.assets/binary_tree_definition.png" /></a></p>
<p align="center"> 图 7-1 &nbsp; 父节点、子节点、子树 </p>
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<h2 id="711">7.1.1 &nbsp; 二叉树常见术语<a class="headerlink" href="#711" title="Permanent link">&para;</a></h2>
<p>二叉树的常用术语如图 7-2 所示。</p>
<ul>
<li>「根节点 root node」：位于二叉树顶层的节点，没有父节点。</li>
<li>「叶节点 leaf node」：没有子节点的节点，其两个指针均指向 <code>None</code> 。</li>
<li>「边 edge」：连接两个节点的线段，即节点引用（指针）。</li>
<li>节点所在的「层 level」：从顶至底递增，根节点所在层为 1 。</li>
<li>节点的「度 degree」：节点的子节点的数量。在二叉树中，度的取值范围是 0、1、2 。</li>
<li>二叉树的「高度 height」：从根节点到最远叶节点所经过的边的数量。</li>
<li>节点的「深度 depth」：从根节点到该节点所经过的边的数量。</li>
<li>节点的「高度 height」：从距离该节点最远的叶节点到该节点所经过的边的数量。</li>
<li><u>根节点（root node）</u>：位于二叉树顶层的节点，没有父节点。</li>
<li><u>叶节点（leaf node）</u>：没有子节点的节点，其两个指针均指向 <code>None</code> 。</li>
<li><u>边（edge）</u>：连接两个节点的线段，即节点引用（指针）。</li>
<li>节点所在的<u>层（level）</u>：从顶至底递增，根节点所在层为 1 。</li>
<li>节点的<u>度（degree）</u>：节点的子节点的数量。在二叉树中，度的取值范围是 0、1、2 。</li>
<li>二叉树的<u>高度（height）</u>：从根节点到最远叶节点所经过的边的数量。</li>
<li>节点的<u>深度（depth）</u>：从根节点到该节点所经过的边的数量。</li>
<li>节点的<u>高度（height）</u>：从距离该节点最远的叶节点到该节点所经过的边的数量。</li>
</ul>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_terminology.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉树的常用术语" class="animation-figure" src="../binary_tree.assets/binary_tree_terminology.png" /></a></p>
<p align="center"> 图 7-2 &nbsp; 二叉树的常用术语 </p>
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<h2 id="713">7.1.3 &nbsp; 常见二叉树类型<a class="headerlink" href="#713" title="Permanent link">&para;</a></h2>
<h3 id="1_1">1. &nbsp; 完美二叉树<a class="headerlink" href="#1_1" title="Permanent link">&para;</a></h3>
<p>如图 7-4 所示，「完美二叉树 perfect binary tree」所有层的节点都被完全填满。在完美二叉树中，叶节点的度为 <span class="arithmatex">\(0\)</span> ，其余所有节点的度都为 <span class="arithmatex">\(2\)</span> ；若树的高度为 <span class="arithmatex">\(h\)</span> ，则节点总数为 <span class="arithmatex">\(2^{h+1} - 1\)</span> ，呈现标准的指数级关系，反映了自然界中常见的细胞分裂现象。</p>
<p>如图 7-4 所示，<u>完美二叉树（perfect binary tree）</u>所有层的节点都被完全填满。在完美二叉树中，叶节点的度为 <span class="arithmatex">\(0\)</span> ，其余所有节点的度都为 <span class="arithmatex">\(2\)</span> ；若树的高度为 <span class="arithmatex">\(h\)</span> ，则节点总数为 <span class="arithmatex">\(2^{h+1} - 1\)</span> ，呈现标准的指数级关系，反映了自然界中常见的细胞分裂现象。</p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>请注意，在中文社区中，完美二叉树常被称为「满二叉树」。</p>
<p>请注意，在中文社区中，完美二叉树常被称为<u>满二叉树</u>。</p>
</div>
<p><a class="glightbox" href="../binary_tree.assets/perfect_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完美二叉树" class="animation-figure" src="../binary_tree.assets/perfect_binary_tree.png" /></a></p>
<p align="center"> 图 7-4 &nbsp; 完美二叉树 </p>
<h3 id="2_1">2. &nbsp; 完全二叉树<a class="headerlink" href="#2_1" title="Permanent link">&para;</a></h3>
<p>如图 7-5 所示，「完全二叉树 complete binary tree」只有最底层的节点未被填满，且最底层节点尽量靠左填充。</p>
<p>如图 7-5 所示，<u>完全二叉树（complete binary tree）</u>只有最底层的节点未被填满，且最底层节点尽量靠左填充。</p>
<p><a class="glightbox" href="../binary_tree.assets/complete_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完全二叉树" class="animation-figure" src="../binary_tree.assets/complete_binary_tree.png" /></a></p>
<p align="center"> 图 7-5 &nbsp; 完全二叉树 </p>
<h3 id="3">3. &nbsp; 完满二叉树<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
<p>如图 7-6 所示，「完满二叉树 full binary tree」除了叶节点之外，其余所有节点都有两个子节点。</p>
<p>如图 7-6 所示，<u>完满二叉树（full binary tree）</u>除了叶节点之外，其余所有节点都有两个子节点。</p>
<p><a class="glightbox" href="../binary_tree.assets/full_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完满二叉树" class="animation-figure" src="../binary_tree.assets/full_binary_tree.png" /></a></p>
<p align="center"> 图 7-6 &nbsp; 完满二叉树 </p>
<h3 id="4">4. &nbsp; 平衡二叉树<a class="headerlink" href="#4" title="Permanent link">&para;</a></h3>
<p>如图 7-7 所示，「平衡二叉树 balanced binary tree」中任意节点的左子树和右子树的高度之差的绝对值不超过 1 。</p>
<p>如图 7-7 所示，<u>平衡二叉树（balanced binary tree）</u>中任意节点的左子树和右子树的高度之差的绝对值不超过 1 。</p>
<p><a class="glightbox" href="../binary_tree.assets/balanced_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="平衡二叉树" class="animation-figure" src="../binary_tree.assets/balanced_binary_tree.png" /></a></p>
<p align="center"> 图 7-7 &nbsp; 平衡二叉树 </p>
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第 4 章 &nbsp; 数组与链表
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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<a href="../../chapter_paperbook/" class="md-nav__link ">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M96 0C43 0 0 43 0 96v320c0 53 43 96 96 96h320c17.7 0 32-14.3 32-32s-14.3-32-32-32v-64c17.7 0 32-14.3 32-32V32c0-17.7-14.3-32-32-32H96zm0 384h256v64H96c-17.7 0-32-14.3-32-32s14.3-32 32-32zm32-240c0-8.8 7.2-16 16-16h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16zm16 48h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16s7.2-16 16-16z"/></svg>
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<p>从物理结构的角度来看，树是一种基于链表的数据结构，因此其遍历方式是通过指针逐个访问节点。然而，树是一种非线性数据结构，这使得遍历树比遍历链表更加复杂，需要借助搜索算法来实现。</p>
<p>二叉树常见的遍历方式包括层序遍历、前序遍历、中序遍历和后序遍历等。</p>
<h2 id="721">7.2.1 &nbsp; 层序遍历<a class="headerlink" href="#721" title="Permanent link">&para;</a></h2>
<p>如图 7-9 所示，「层序遍历 level-order traversal」从顶部到底部逐层遍历二叉树，并在每一层按照从左到右的顺序访问节点。</p>
<p>层序遍历本质上属于「广度优先遍历 breadth-first traversal」，也称「广度优先搜索 breadth-first search, BFS」，它体现了一种“一圈一圈向外扩展”的逐层遍历方式。</p>
<p>如图 7-9 所示，<u>层序遍历（level-order traversal）</u>从顶部到底部逐层遍历二叉树，并在每一层按照从左到右的顺序访问节点。</p>
<p>层序遍历本质上属于<u>广度优先遍历（breadth-first traversal）</u>，也称<u>广度优先搜索（breadth-first search, BFS）</u>，它体现了一种“一圈一圈向外扩展”的逐层遍历方式。</p>
<p><a class="glightbox" href="../binary_tree_traversal.assets/binary_tree_bfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉树的层序遍历" class="animation-figure" src="../binary_tree_traversal.assets/binary_tree_bfs.png" /></a></p>
<p align="center"> 图 7-9 &nbsp; 二叉树的层序遍历 </p>
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<li><strong>空间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> ：在最差情况下，即满二叉树时，遍历到最底层之前，队列中最多同时存在 <span class="arithmatex">\((n + 1) / 2\)</span> 个节点，占用 <span class="arithmatex">\(O(n)\)</span> 空间。</li>
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<h2 id="722">7.2.2 &nbsp; 前序、中序、后序遍历<a class="headerlink" href="#722" title="Permanent link">&para;</a></h2>
<p>相应地，前序、中序和后序遍历都属于「深度优先遍历 depth-first traversal」，也称「深度优先搜索 depth-first search, DFS」，它体现了一种“先走到尽头，再回溯继续”的遍历方式。</p>
<p>相应地，前序、中序和后序遍历都属于<u>深度优先遍历（depth-first traversal）</u>，也称<u>深度优先搜索（depth-first search, DFS）</u>，它体现了一种“先走到尽头，再回溯继续”的遍历方式。</p>
<p>图 7-10 展示了对二叉树进行深度优先遍历的工作原理。<strong>深度优先遍历就像是绕着整棵二叉树的外围“走”一圈</strong>，在每个节点都会遇到三个位置，分别对应前序遍历、中序遍历和后序遍历。</p>
<p><a class="glightbox" href="../binary_tree_traversal.assets/binary_tree_dfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树的前序、中序、后序遍历" class="animation-figure" src="../binary_tree_traversal.assets/binary_tree_dfs.png" /></a></p>
<p align="center"> 图 7-10 &nbsp; 二叉搜索树的前序、中序、后序遍历 </p>
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 10 章 &nbsp; 搜索
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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纸质书
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纸质书
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Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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chapter_tree/summary/index.html
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序
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 10 章 &nbsp; 搜索
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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纸质书
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纸质书
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