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<h2 id="731">7.3.1 &nbsp; 表示完美二叉树<a class="headerlink" href="#731" title="Permanent link">&para;</a></h2>
<p>先分析一个简单案例。给定一个完美二叉树，我们将所有节点按照层序遍历的顺序存储在一个数组中，则每个节点都对应唯一的数组索引。</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> 。图 7-12 展示了各个节点索引之间的映射关系。</p>
<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>
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完美二叉树的数组表示" src="../array_representation_of_tree.assets/array_representation_binary_tree.png" /></a></p>
<p align="center"> 图 7-12 &nbsp; 完美二叉树的数组表示 </p>
<p><strong>映射公式的角色相当于链表中的指针</strong>。给定数组中的任意一个节点，我们都可以通过映射公式来访问它的左（右）子节点。</p>
<h2 id="732">7.3.2 &nbsp; 表示任意二叉树<a class="headerlink" href="#732" title="Permanent link">&para;</a></h2>
<p>完美二叉树是一个特例，在二叉树的中间层通常存在许多 <span class="arithmatex">\(\text{None}\)</span> 。由于层序遍历序列并不包含这些 <span class="arithmatex">\(\text{None}\)</span> ，因此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>。</p>
<p>如图 7-13 所示，给定一个非完美二叉树，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>
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_without_empty.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><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" /></a></p>
<p align="center"> 图 7-13 &nbsp; 层序遍历序列对应多种二叉树可能性 </p>
<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> 。如图 7-14 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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<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>
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_with_empty.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><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" /></a></p>
<p align="center"> 图 7-14 &nbsp; 任意类型二叉树的数组表示 </p>
<p>值得说明的是，<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>因此所有 <span class="arithmatex">\(\text{None}\)</span> 一定出现在层序遍历序列的末尾</strong>。</p>
<p>这意味着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> ，非常方便。图 7-15 给出了一个例子。</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>
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完全二叉树的数组表示" src="../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" /></a></p>
<p align="center"> 图 7-15 &nbsp; 完全二叉树的数组表示 </p>
<p>以下代码实现了一个基于数组表示的二叉树，包括以下几种操作。</p>
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<h1 id="75-avl">7.5 &nbsp; AVL 树 *<a class="headerlink" href="#75-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>如图 7-24 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>
<p><a class="glightbox" href="../avl_tree.assets/avltree_degradation_from_removing_node.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="AVL 树在删除节点后发生退化" src="../avl_tree.assets/avltree_degradation_from_removing_node.png" /></a></p>
<p align="center"> 图 7-24 &nbsp; AVL 树在删除节点后发生退化 </p>
<p>再例如，在图 7-25 的完美二叉树中插入两个节点后，树将严重向左倾斜，查找操作的时间复杂度也随之恶化。</p>
<p><img alt="AVL 树在插入节点后发生退化" src="../avl_tree.assets/avltree_degradation_from_inserting_node.png" /></p>
<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" 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>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>
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<p align="center"> 图 7-26 &nbsp; 右旋操作步骤 </p>
<p>如图 7-27 所示，当节点 <code>child</code> 有右子节点（记为 <code>grandChild</code> ）时，需要在右旋中添加一步：将 <code>grandChild</code> 作为 <code>node</code> 的左子节点。</p>
<p><img alt="有 grandChild 的右旋操作" src="../avl_tree.assets/avltree_right_rotate_with_grandchild.png" /></p>
<p><a class="glightbox" href="../avl_tree.assets/avltree_right_rotate_with_grandchild.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有 grandChild 的右旋操作" src="../avl_tree.assets/avltree_right_rotate_with_grandchild.png" /></a></p>
<p align="center"> 图 7-27 &nbsp; 有 grandChild 的右旋操作 </p>
<p>“向右旋转”是一种形象化的说法，实际上需要通过修改节点指针来实现，代码如下所示。</p>
@@ -4412,11 +4247,11 @@
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<p>相应的，如果考虑上述失衡二叉树的“镜像”，则需要执行图 7-28 所示的“左旋”操作。</p>
<p><img alt="左旋操作" src="../avl_tree.assets/avltree_left_rotate.png" /></p>
<p><a class="glightbox" href="../avl_tree.assets/avltree_left_rotate.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="左旋操作" src="../avl_tree.assets/avltree_left_rotate.png" /></a></p>
<p align="center"> 图 7-28 &nbsp; 左旋操作 </p>
<p>同理，如图 7-29 所示，当节点 <code>child</code> 有左子节点（记为 <code>grandChild</code> ）时，需要在左旋中添加一步：将 <code>grandChild</code> 作为 <code>node</code> 的右子节点。</p>
<p><img alt="有 grandChild 的左旋操作" src="../avl_tree.assets/avltree_left_rotate_with_grandchild.png" /></p>
<p><a class="glightbox" href="../avl_tree.assets/avltree_left_rotate_with_grandchild.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有 grandChild 的左旋操作" src="../avl_tree.assets/avltree_left_rotate_with_grandchild.png" /></a></p>
<p align="center"> 图 7-29 &nbsp; 有 grandChild 的左旋操作 </p>
<p>可以观察到，<strong>右旋和左旋操作在逻辑上是镜像对称的，它们分别解决的两种失衡情况也是对称的</strong>。基于对称性，我们只需将右旋的实现代码中的所有的 <code>left</code> 替换为 <code>right</code> ，将所有的 <code>right</code> 替换为 <code>left</code> ，即可得到左旋的实现代码。</p>
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<h3 id="3">3. &nbsp; 先左旋后右旋<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
<p>对于图 7-30 中的失衡节点 3 ，仅使用左旋或右旋都无法使子树恢复平衡。此时需要先对 <code>child</code> 执行“左旋”，再对 <code>node</code> 执行“右旋”。</p>
<p><img alt="先左旋后右旋" src="../avl_tree.assets/avltree_left_right_rotate.png" /></p>
<p><a class="glightbox" href="../avl_tree.assets/avltree_left_right_rotate.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="先左旋后右旋" src="../avl_tree.assets/avltree_left_right_rotate.png" /></a></p>
<p align="center"> 图 7-30 &nbsp; 先左旋后右旋 </p>
<h3 id="4">4. &nbsp; 先右旋后左旋<a class="headerlink" href="#4" title="Permanent link">&para;</a></h3>
<p>如图 7-31 所示，对于上述失衡二叉树的镜像情况，需要先对 <code>child</code> 执行“右旋”，然后对 <code>node</code> 执行“左旋”。</p>
<p><img alt="先右旋后左旋" src="../avl_tree.assets/avltree_right_left_rotate.png" /></p>
<p><a class="glightbox" href="../avl_tree.assets/avltree_right_left_rotate.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="先右旋后左旋" src="../avl_tree.assets/avltree_right_left_rotate.png" /></a></p>
<p align="center"> 图 7-31 &nbsp; 先右旋后左旋 </p>
<h3 id="5">5. &nbsp; 旋转的选择<a class="headerlink" href="#5" title="Permanent link">&para;</a></h3>
<p>图 7-32 展示的四种失衡情况与上述案例逐个对应，分别需要采用右旋、左旋、先右后左、先左后右的旋转操作。</p>
<p><img alt="AVL 树的四种旋转情况" src="../avl_tree.assets/avltree_rotation_cases.png" /></p>
<p><a class="glightbox" href="../avl_tree.assets/avltree_rotation_cases.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="AVL 树的四种旋转情况" src="../avl_tree.assets/avltree_rotation_cases.png" /></a></p>
<p align="center"> 图 7-32 &nbsp; AVL 树的四种旋转情况 </p>
<p>如下表所示，我们通过判断失衡节点的平衡因子以及较高一侧子节点的平衡因子的正负号，来确定失衡节点属于图 7-32 中的哪种情况。</p>
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10.2 &nbsp; 二分查找插入点
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<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>
</ol>
<p><img alt="二叉搜索树" src="../binary_search_tree.assets/binary_search_tree.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/binary_search_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树" src="../binary_search_tree.assets/binary_search_tree.png" /></a></p>
<p align="center"> 图 7-16 &nbsp; 二叉搜索树 </p>
<h2 id="741">7.4.1 &nbsp; 二叉搜索树的操作<a class="headerlink" href="#741" title="Permanent link">&para;</a></h2>
@@ -3576,16 +3411,16 @@
<div class="tabbed-set tabbed-alternate" data-tabs="1:4"><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" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label></div>
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<p><img alt="二叉搜索树查找节点示例" src="../binary_search_tree.assets/bst_search_step1.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_search_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树查找节点示例" src="../binary_search_tree.assets/bst_search_step1.png" /></a></p>
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<p><img alt="bst_search_step2" src="../binary_search_tree.assets/bst_search_step2.png" /></p>
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<p><img alt="bst_search_step3" src="../binary_search_tree.assets/bst_search_step3.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_search_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="bst_search_step3" src="../binary_search_tree.assets/bst_search_step3.png" /></a></p>
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<p><img alt="bst_search_step4" src="../binary_search_tree.assets/bst_search_step4.png" /></p>
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<li><strong>查找插入位置</strong>：与查找操作相似，从根节点出发，根据当前节点值和 <code>num</code> 的大小关系循环向下搜索，直到越过叶节点（遍历至 <span class="arithmatex">\(\text{None}\)</span> ）时跳出循环。</li>
<li><strong>在该位置插入节点</strong>：初始化节点 <code>num</code> ，将该节点置于 <span class="arithmatex">\(\text{None}\)</span> 的位置。</li>
</ol>
<p><img alt="在二叉搜索树中插入节点" src="../binary_search_tree.assets/bst_insert.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_insert.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉搜索树中插入节点" src="../binary_search_tree.assets/bst_insert.png" /></a></p>
<p align="center"> 图 7-18 &nbsp; 在二叉搜索树中插入节点 </p>
<p>在代码实现中，需要注意以下两点。</p>
@@ -4256,11 +4091,11 @@
<p>与插入节点类似，我们需要保证在删除操作完成后，二叉搜索树的“左子树 &lt; 根节点 &lt; 右子树”的性质仍然满足。</p>
<p>因此，我们需要根据目标节点的子节点数量，共分为 0、1 和 2 这三种情况，执行对应的删除节点操作。</p>
<p>如图 7-19 所示，当待删除节点的度为 <span class="arithmatex">\(0\)</span> 时，表示该节点是叶节点，可以直接删除。</p>
<p><img alt="在二叉搜索树中删除节点（度为 0 ）" src="../binary_search_tree.assets/bst_remove_case1.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉搜索树中删除节点（度为 0 ）" src="../binary_search_tree.assets/bst_remove_case1.png" /></a></p>
<p align="center"> 图 7-19 &nbsp; 在二叉搜索树中删除节点（度为 0 ） </p>
<p>如图 7-20 所示，当待删除节点的度为 <span class="arithmatex">\(1\)</span> 时，将待删除节点替换为其子节点即可。</p>
<p><img alt="在二叉搜索树中删除节点（度为 1 ）" src="../binary_search_tree.assets/bst_remove_case2.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉搜索树中删除节点（度为 1 ）" src="../binary_search_tree.assets/bst_remove_case2.png" /></a></p>
<p align="center"> 图 7-20 &nbsp; 在二叉搜索树中删除节点（度为 1 ） </p>
<p>当待删除节点的度为 <span class="arithmatex">\(2\)</span> 时，我们无法直接删除它，而需要使用一个节点替换该节点。由于要保持二叉搜索树“左 <span class="arithmatex">\(&lt;\)</span> 根 <span class="arithmatex">\(&lt;\)</span> 右”的性质，<strong>因此这个节点可以是右子树的最小节点或左子树的最大节点</strong>。</p>
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@@ -4942,7 +4777,7 @@
<p>如图 7-22 所示，二叉树的中序遍历遵循“左 <span class="arithmatex">\(\rightarrow\)</span> 根 <span class="arithmatex">\(\rightarrow\)</span> 右”的遍历顺序，而二叉搜索树满足“左子节点 <span class="arithmatex">\(&lt;\)</span> 根节点 <span class="arithmatex">\(&lt;\)</span> 右子节点”的大小关系。</p>
<p>这意味着在二叉搜索树中进行中序遍历时，总是会优先遍历下一个最小节点，从而得出一个重要性质：<strong>二叉搜索树的中序遍历序列是升序的</strong>。</p>
<p>利用中序遍历升序的性质，我们在二叉搜索树中获取有序数据仅需 <span class="arithmatex">\(O(n)\)</span> 时间，无须进行额外的排序操作，非常高效。</p>
<p><img alt="二叉搜索树的中序遍历序列" src="../binary_search_tree.assets/bst_inorder_traversal.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_inorder_traversal.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树的中序遍历序列" src="../binary_search_tree.assets/bst_inorder_traversal.png" /></a></p>
<p align="center"> 图 7-22 &nbsp; 二叉搜索树的中序遍历序列 </p>
<h2 id="742">7.4.2 &nbsp; 二叉搜索树的效率<a class="headerlink" href="#742" title="Permanent link">&para;</a></h2>
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<p>在理想情况下，二叉搜索树是“平衡”的，这样就可以在 <span class="arithmatex">\(\log n\)</span> 轮循环内查找任意节点。</p>
<p>然而，如果我们在二叉搜索树中不断地插入和删除节点，可能导致二叉树退化为图 7-23 所示的链表，这时各种操作的时间复杂度也会退化为 <span class="arithmatex">\(O(n)\)</span> 。</p>
<p><img alt="二叉搜索树的退化" src="../binary_search_tree.assets/bst_degradation.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_degradation.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树的退化" src="../binary_search_tree.assets/bst_degradation.png" /></a></p>
<p align="center"> 图 7-23 &nbsp; 二叉搜索树的退化 </p>
<h2 id="743">7.4.3 &nbsp; 二叉搜索树常见应用<a class="headerlink" href="#743" title="Permanent link">&para;</a></h2>
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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><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><img alt="父节点、子节点、子树" src="../binary_tree.assets/binary_tree_definition.png" /></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" 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>
<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>
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<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」：从最远叶节点到该节点所经过的边的数量。</li>
</ul>
<p><img alt="二叉树的常用术语" src="../binary_tree.assets/binary_tree_terminology.png" /></p>
<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" 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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<h3 id="2">2. &nbsp; 插入与删除节点<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p>与链表类似，在二叉树中插入与删除节点可以通过修改指针来实现。图 7-3 给出了一个示例。</p>
<p><img alt="在二叉树中插入与删除节点" src="../binary_tree.assets/binary_tree_add_remove.png" /></p>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_add_remove.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉树中插入与删除节点" src="../binary_tree.assets/binary_tree_add_remove.png" /></a></p>
<p align="center"> 图 7-3 &nbsp; 在二叉树中插入与删除节点 </p>
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Zig</label></div>
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<p>请注意，在中文社区中，完美二叉树常被称为「满二叉树」。</p>
</div>
<p><img alt="完美二叉树" src="../binary_tree.assets/perfect_binary_tree.png" /></p>
<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" 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><img alt="完全二叉树" src="../binary_tree.assets/complete_binary_tree.png" /></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" 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><img alt="完满二叉树" src="../binary_tree.assets/full_binary_tree.png" /></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" 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><img alt="平衡二叉树" src="../binary_tree.assets/balanced_binary_tree.png" /></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" src="../binary_tree.assets/balanced_binary_tree.png" /></a></p>
<p align="center"> 图 7-7 &nbsp; 平衡二叉树 </p>
<h2 id="714">7.1.4 &nbsp; 二叉树的退化<a class="headerlink" href="#714" title="Permanent link">&para;</a></h2>
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<li>完美二叉树是理想情况，可以充分发挥二叉树“分治”的优势。</li>
<li>链表则是另一个极端，各项操作都变为线性操作，时间复杂度退化至 <span class="arithmatex">\(O(n)\)</span> 。</li>
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<p><img alt="二叉树的最佳与最差结构" src="../binary_tree.assets/binary_tree_best_worst_cases.png" /></p>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_best_worst_cases.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉树的最佳与最差结构" src="../binary_tree.assets/binary_tree_best_worst_cases.png" /></a></p>
<p align="center"> 图 7-8 &nbsp; 二叉树的最佳与最差结构 </p>
<p>如表 7-1 所示，在最佳和最差结构下，二叉树的叶节点数量、节点总数、高度等达到极大或极小值。</p>
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Copyright &copy; 2023 Krahets
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<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」，它体现了一种“一圈一圈向外扩展”的逐层遍历方式。</p>
<p><img alt="二叉树的层序遍历" src="../binary_tree_traversal.assets/binary_tree_bfs.png" /></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" 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>
<h3 id="1">1. &nbsp; 代码实现<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
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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」，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个二叉树的外围“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><img alt="二叉搜索树的前、中、后序遍历" src="../binary_tree_traversal.assets/binary_tree_dfs.png" /></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、中、后序遍历" 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、中、后序遍历 </p>
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<div class="md-copyright">
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<h1 id="7">第 7 章 &nbsp; 树<a class="headerlink" href="#7" title="Permanent link">&para;</a></h1>
<div class="center-table">
<p><img alt="树" src="../assets/covers/chapter_tree.jpg" width="600" /></p>
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<div class="md-copyright">
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Copyright &copy; 2023 Krahets
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@@ -3665,5 +3505,5 @@ aria-label="页脚"
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<script>document$.subscribe(() => {const lightbox = GLightbox({"touchNavigation": true, "loop": false, "zoomable": true, "draggable": false, "openEffect": "zoom", "closeEffect": "zoom", "slideEffect": "none"});})</script></body>
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<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
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14.1 &nbsp; 初探动态规划
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15.1 &nbsp; 贪心算法
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15.3 &nbsp; 最大容量问题
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15.4 &nbsp; 最大切分乘积问题
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15.5 &nbsp; 小结
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</li>
@@ -3689,10 +3524,15 @@ aria-label="页脚"
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Copyright &copy; 2023 Krahets
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@@ -3761,5 +3601,5 @@ aria-label="页脚"
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