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2026-07-20 10:36:09 +00:00 · 2023-12-28 17:18:44 +08:00
parent 5e0d7327db
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@@ -1788,7 +1788,7 @@
        
  
  <span class="md-ellipsis">
-    8.3 &nbsp; Top-K 问题
+    8.3 &nbsp; Top-k 问题
  </span>
  

@@ -3543,12 +3543,12 @@

 <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>完美二叉树是一个特例，在二叉树的中间层通常存在许多 <code>None</code> 。由于层序遍历序列并不包含这些 <code>None</code> ，因此我们无法仅凭该序列来推测 <code>None</code> 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，给定一棵非完美二叉树，上述数组表示方法已经失效。</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" class="animation-figure" 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。示例代码如下：</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 <code>None</code></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。示例代码如下：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:12"><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" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -3626,8 +3626,8 @@
 <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" class="animation-figure" 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>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，<code>None</code> 只出现在最底层且靠右的位置，<strong>因此所有 <code>None</code> 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 <code>None</code> ，非常方便。图 7-15 给出了一个例子。</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" class="animation-figure" 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>

@@ -4645,7 +4645,7 @@
 <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>
-<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>
+<li>当二叉树中存在大量 <code>None</code> 时，数组中包含的节点数据比重较低，空间利用率较低。</li>
 </ul>

 <!-- Source file information -->
@@ -1905,7 +1905,7 @@
        
  
  <span class="md-ellipsis">
-    8.3 &nbsp; Top-K 问题
+    8.3 &nbsp; Top-k 问题
  </span>
  

@@ -3767,16 +3767,16 @@

 <!-- Page content -->
 <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，在多次插入和删除操作后，二叉搜索树可能退化为链表。在这种情况下，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>在“二叉搜索树”章节中我们提到，在多次插入和删除操作后，二叉搜索树可能退化为链表。在这种情况下，所有操作的时间复杂度将从 <span class="arithmatex">\(O(\log n)\)</span> 劣化为 <span class="arithmatex">\(O(n)\)</span> 。</p>
 <p>如图 7-24 所示，经过两次删除节点操作，这棵二叉搜索树便会退化为链表。</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" class="animation-figure" 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 所示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>再例如，在图 7-25 所示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劣化。</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" 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「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="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，因此也被称为「平衡二叉搜索树 balanced binary search tree」。</p>
 <h3 id="1">1. &nbsp; 节点高度<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
@@ -3947,7 +3947,7 @@
 </div>
 </div>
 </div>
-<p>“节点高度”是指从该节点到其最远叶节点的距离，即所经过的“边”的数量。需要特别注意的是，叶节点的高度为 <span class="arithmatex">\(0\)</span> ，而空节点的高度为 <span class="arithmatex">\(-1\)</span> 。我们将创建两个工具函数，分别用于获取和更新节点的高度：</p>
+<p>“节点高度”是指从该节点到它的最远叶节点的距离，即所经过的“边”的数量。需要特别注意的是，叶节点的高度为 <span class="arithmatex">\(0\)</span> ，而空节点的高度为 <span class="arithmatex">\(-1\)</span> 。我们将创建两个工具函数，分别用于获取和更新节点的高度：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -4311,9 +4311,9 @@
 </div>
 <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><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" class="animation-figure" 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>如图 7-27 所示，当节点 <code>child</code> 有右子节点（记为 <code>grand_child</code> ）时，需要在右旋中添加一步：将 <code>grand_child</code> 作为 <code>node</code> 的左子节点。</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 grand_child 的右旋操作" class="animation-figure" src="../avl_tree.assets/avltree_right_rotate_with_grandchild.png" /></a></p>
+<p align="center"> 图 7-27 &nbsp; 有 grand_child 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>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Zig</label></div>
@@ -4522,9 +4522,9 @@
 <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" class="animation-figure" 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><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" class="animation-figure" 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>同理，如图 7-29 所示，当节点 <code>child</code> 有左子节点（记为 <code>grand_child</code> ）时，需要在左旋中添加一步：将 <code>grand_child</code> 作为 <code>node</code> 的右子节点。</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 grand_child 的左旋操作" class="animation-figure" src="../avl_tree.assets/avltree_left_rotate_with_grandchild.png" /></a></p>
+<p align="center"> 图 7-29 &nbsp; 有 grand_child 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>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Zig</label></div>
@@ -5183,7 +5183,7 @@
 <a id="__codelineno-72-6" name="__codelineno-72-6" href="#__codelineno-72-6"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;递归插入节点（辅助方法）&quot;&quot;&quot;</span>
 <a id="__codelineno-72-7" name="__codelineno-72-7" href="#__codelineno-72-7"></a>    <span class="k">if</span> <span class="n">node</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-72-8" name="__codelineno-72-8" href="#__codelineno-72-8"></a>        <span class="k">return</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-72-9" name="__codelineno-72-9" href="#__codelineno-72-9"></a>    <span class="c1"># 1. 查找插入位置，并插入节点</span>
+<a id="__codelineno-72-9" name="__codelineno-72-9" href="#__codelineno-72-9"></a>    <span class="c1"># 1. 查找插入位置并插入节点</span>
 <a id="__codelineno-72-10" name="__codelineno-72-10" href="#__codelineno-72-10"></a>    <span class="k">if</span> <span class="n">val</span> <span class="o">&lt;</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
 <a id="__codelineno-72-11" name="__codelineno-72-11" href="#__codelineno-72-11"></a>        <span class="n">node</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">insert_helper</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-72-12" name="__codelineno-72-12" href="#__codelineno-72-12"></a>    <span class="k">elif</span> <span class="n">val</span> <span class="o">&gt;</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
@@ -5207,7 +5207,7 @@
 <a id="__codelineno-73-7" name="__codelineno-73-7" href="#__codelineno-73-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">insertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-73-8" name="__codelineno-73-8" href="#__codelineno-73-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span>
 <a id="__codelineno-73-9" name="__codelineno-73-9" href="#__codelineno-73-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-73-10" name="__codelineno-73-10" href="#__codelineno-73-10"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-73-10" name="__codelineno-73-10" href="#__codelineno-73-10"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-73-11" name="__codelineno-73-11" href="#__codelineno-73-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-73-12" name="__codelineno-73-12" href="#__codelineno-73-12"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-73-13" name="__codelineno-73-13" href="#__codelineno-73-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span>
@@ -5232,7 +5232,7 @@
 <a id="__codelineno-74-7" name="__codelineno-74-7" href="#__codelineno-74-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">insertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-8" name="__codelineno-74-8" href="#__codelineno-74-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-74-9" name="__codelineno-74-9" href="#__codelineno-74-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-74-10" name="__codelineno-74-10" href="#__codelineno-74-10"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-74-10" name="__codelineno-74-10" href="#__codelineno-74-10"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-74-11" name="__codelineno-74-11" href="#__codelineno-74-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span>
 <a id="__codelineno-74-12" name="__codelineno-74-12" href="#__codelineno-74-12"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-74-13" name="__codelineno-74-13" href="#__codelineno-74-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span>
@@ -5256,7 +5256,7 @@
 <a id="__codelineno-75-6" name="__codelineno-75-6" href="#__codelineno-75-6"></a><span class="cm">/* 递归插入节点（辅助方法） */</span>
 <a id="__codelineno-75-7" name="__codelineno-75-7" href="#__codelineno-75-7"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">InsertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-75-8" name="__codelineno-75-8" href="#__codelineno-75-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-75-9" name="__codelineno-75-9" href="#__codelineno-75-9"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-75-9" name="__codelineno-75-9" href="#__codelineno-75-9"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-75-10" name="__codelineno-75-10" href="#__codelineno-75-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-75-11" name="__codelineno-75-11" href="#__codelineno-75-11"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">InsertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-75-12" name="__codelineno-75-12" href="#__codelineno-75-12"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
@@ -5282,7 +5282,7 @@
 <a id="__codelineno-76-8" name="__codelineno-76-8" href="#__codelineno-76-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-9" name="__codelineno-76-9" href="#__codelineno-76-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-76-10" name="__codelineno-76-10" href="#__codelineno-76-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-76-11" name="__codelineno-76-11" href="#__codelineno-76-11"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-76-11" name="__codelineno-76-11" href="#__codelineno-76-11"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-76-12" name="__codelineno-76-12" href="#__codelineno-76-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-13" name="__codelineno-76-13" href="#__codelineno-76-13"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-76-14" name="__codelineno-76-14" href="#__codelineno-76-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5312,7 +5312,7 @@
 <a id="__codelineno-77-9" name="__codelineno-77-9" href="#__codelineno-77-9"></a>    <span class="k">if</span> <span class="n">node</span> <span class="p">==</span> <span class="kc">nil</span> <span class="p">{</span>
 <a id="__codelineno-77-10" name="__codelineno-77-10" href="#__codelineno-77-10"></a>        <span class="k">return</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-77-11" name="__codelineno-77-11" href="#__codelineno-77-11"></a>    <span class="p">}</span>
-<a id="__codelineno-77-12" name="__codelineno-77-12" href="#__codelineno-77-12"></a>    <span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-77-12" name="__codelineno-77-12" href="#__codelineno-77-12"></a>    <span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-77-13" name="__codelineno-77-13" href="#__codelineno-77-13"></a>    <span class="k">if</span> <span class="n">val</span> <span class="o">&lt;</span> <span class="n">node</span><span class="p">!.</span><span class="n">val</span> <span class="p">{</span>
 <a id="__codelineno-77-14" name="__codelineno-77-14" href="#__codelineno-77-14"></a>        <span class="n">node</span><span class="p">?.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span> <span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-77-15" name="__codelineno-77-15" href="#__codelineno-77-15"></a>    <span class="p">}</span> <span class="k">else</span> <span class="k">if</span> <span class="n">val</span> <span class="o">&gt;</span> <span class="n">node</span><span class="p">!.</span><span class="n">val</span> <span class="p">{</span>
@@ -5337,7 +5337,7 @@
 <a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a><span class="cm">/* 递归插入节点（辅助方法） */</span>
 <a id="__codelineno-78-7" name="__codelineno-78-7" href="#__codelineno-78-7"></a><span class="err">#</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-78-8" name="__codelineno-78-8" href="#__codelineno-78-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</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="k">return</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
-<a id="__codelineno-78-9" name="__codelineno-78-9" href="#__codelineno-78-9"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-78-9" name="__codelineno-78-9" href="#__codelineno-78-9"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-78-10" name="__codelineno-78-10" href="#__codelineno-78-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-78-11" name="__codelineno-78-11" href="#__codelineno-78-11"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-78-12" name="__codelineno-78-12" href="#__codelineno-78-12"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
@@ -5359,7 +5359,7 @@
 <a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="cm">/* 递归插入节点（辅助方法） */</span>
 <a id="__codelineno-79-7" name="__codelineno-79-7" href="#__codelineno-79-7"></a><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-8" name="__codelineno-79-8" href="#__codelineno-79-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</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="k">return</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
-<a id="__codelineno-79-9" name="__codelineno-79-9" href="#__codelineno-79-9"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-79-9" name="__codelineno-79-9" href="#__codelineno-79-9"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-79-10" name="__codelineno-79-10" href="#__codelineno-79-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-11" name="__codelineno-79-11" href="#__codelineno-79-11"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-79-12" name="__codelineno-79-12" href="#__codelineno-79-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5384,7 +5384,7 @@
 <a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="cm">/* 递归插入节点（辅助方法） */</span>
 <a id="__codelineno-80-7" name="__codelineno-80-7" href="#__codelineno-80-7"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-8" name="__codelineno-80-8" href="#__codelineno-80-8"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</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="k">return</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-80-9" name="__codelineno-80-9" href="#__codelineno-80-9"></a><span class="w">  </span><span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-80-9" name="__codelineno-80-9" href="#__codelineno-80-9"></a><span class="w">  </span><span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-80-10" name="__codelineno-80-10" href="#__codelineno-80-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-80-11" name="__codelineno-80-11" href="#__codelineno-80-11"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-80-12" name="__codelineno-80-12" href="#__codelineno-80-12"></a><span class="w">  </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
@@ -5409,7 +5409,7 @@
 <a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a><span class="k">fn</span> <span class="nf">insert_helper</span><span class="p">(</span><span class="n">node</span>: <span class="nc">OptionTreeNodeRc</span><span class="p">,</span><span class="w"> </span><span class="n">val</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="nc">OptionTreeNodeRc</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></a><span class="w">    </span><span class="k">match</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-9" name="__codelineno-81-9" href="#__codelineno-81-9"></a><span class="w">        </span><span class="nb">Some</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-81-10" name="__codelineno-81-10" href="#__codelineno-81-10"></a><span class="w">            </span><span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-81-10" name="__codelineno-81-10" href="#__codelineno-81-10"></a><span class="w">            </span><span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-81-11" name="__codelineno-81-11" href="#__codelineno-81-11"></a><span class="w">            </span><span class="k">match</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-12" name="__codelineno-81-12" href="#__codelineno-81-12"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="n">node_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-81-13" name="__codelineno-81-13" href="#__codelineno-81-13"></a><span class="w">                </span><span class="n">node_val</span>
@@ -5450,7 +5450,7 @@
 <a id="__codelineno-82-8" name="__codelineno-82-8" href="#__codelineno-82-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-9" name="__codelineno-82-9" href="#__codelineno-82-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-82-10" name="__codelineno-82-10" href="#__codelineno-82-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置，并插入节点 */</span>
+<a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="w">    </span><span class="cm">/* 1. 查找插入位置并插入节点 */</span>
 <a id="__codelineno-82-12" name="__codelineno-82-12" href="#__codelineno-82-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-13" name="__codelineno-82-13" href="#__codelineno-82-13"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-82-14" name="__codelineno-82-14" href="#__codelineno-82-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5482,7 +5482,7 @@
 <a id="__codelineno-83-11" name="__codelineno-83-11" href="#__codelineno-83-11"></a><span class="w">        </span><span class="n">tmp_node</span><span class="p">.</span><span class="n">init</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-83-12" name="__codelineno-83-12" href="#__codelineno-83-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tmp_node</span><span class="p">;</span>
 <a id="__codelineno-83-13" name="__codelineno-83-13" href="#__codelineno-83-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-83-14" name="__codelineno-83-14" href="#__codelineno-83-14"></a><span class="w">    </span><span class="c1">// 1. 查找插入位置，并插入节点</span>
+<a id="__codelineno-83-14" name="__codelineno-83-14" href="#__codelineno-83-14"></a><span class="w">    </span><span class="c1">// 1. 查找插入位置并插入节点</span>
 <a id="__codelineno-83-15" name="__codelineno-83-15" href="#__codelineno-83-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-16" name="__codelineno-83-16" href="#__codelineno-83-16"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">try</span><span class="w"> </span><span class="n">self</span><span class="p">.</span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-83-17" name="__codelineno-83-17" href="#__codelineno-83-17"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5513,7 +5513,7 @@
 <a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;递归删除节点（辅助方法）&quot;&quot;&quot;</span>
 <a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a>    <span class="k">if</span> <span class="n">node</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-84-8" name="__codelineno-84-8" href="#__codelineno-84-8"></a>        <span class="k">return</span> <span class="kc">None</span>
-<a id="__codelineno-84-9" name="__codelineno-84-9" href="#__codelineno-84-9"></a>    <span class="c1"># 1. 查找节点，并删除之</span>
+<a id="__codelineno-84-9" name="__codelineno-84-9" href="#__codelineno-84-9"></a>    <span class="c1"># 1. 查找节点并删除</span>
 <a id="__codelineno-84-10" name="__codelineno-84-10" href="#__codelineno-84-10"></a>    <span class="k">if</span> <span class="n">val</span> <span class="o">&lt;</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
 <a id="__codelineno-84-11" name="__codelineno-84-11" href="#__codelineno-84-11"></a>        <span class="n">node</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">remove_helper</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-84-12" name="__codelineno-84-12" href="#__codelineno-84-12"></a>    <span class="k">elif</span> <span class="n">val</span> <span class="o">&gt;</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
@@ -5550,7 +5550,7 @@
 <a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">removeHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-8" name="__codelineno-85-8" href="#__codelineno-85-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span>
 <a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">nullptr</span><span class="p">;</span>
-<a id="__codelineno-85-10" name="__codelineno-85-10" href="#__codelineno-85-10"></a><span class="w">    </span><span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-85-10" name="__codelineno-85-10" href="#__codelineno-85-10"></a><span class="w">    </span><span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-85-11" name="__codelineno-85-11" href="#__codelineno-85-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-85-12" name="__codelineno-85-12" href="#__codelineno-85-12"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-85-13" name="__codelineno-85-13" href="#__codelineno-85-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span>
@@ -5597,7 +5597,7 @@
 <a id="__codelineno-86-7" name="__codelineno-86-7" href="#__codelineno-86-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">removeHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-8" name="__codelineno-86-8" href="#__codelineno-86-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-86-9" name="__codelineno-86-9" href="#__codelineno-86-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-86-10" name="__codelineno-86-10" href="#__codelineno-86-10"></a><span class="w">    </span><span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-86-10" name="__codelineno-86-10" href="#__codelineno-86-10"></a><span class="w">    </span><span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-86-11" name="__codelineno-86-11" href="#__codelineno-86-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span>
 <a id="__codelineno-86-12" name="__codelineno-86-12" href="#__codelineno-86-12"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-86-13" name="__codelineno-86-13" href="#__codelineno-86-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span>
@@ -5638,7 +5638,7 @@
 <a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="cm">/* 递归删除节点（辅助方法） */</span>
 <a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">RemoveHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-8" name="__codelineno-87-8" href="#__codelineno-87-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
-<a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a><span class="w">    </span><span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a><span class="w">    </span><span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-87-10" name="__codelineno-87-10" href="#__codelineno-87-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-87-11" name="__codelineno-87-11" href="#__codelineno-87-11"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">RemoveHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-87-12" name="__codelineno-87-12" href="#__codelineno-87-12"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
@@ -5681,7 +5681,7 @@
 <a id="__codelineno-88-8" name="__codelineno-88-8" href="#__codelineno-88-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-9" name="__codelineno-88-9" href="#__codelineno-88-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-88-10" name="__codelineno-88-10" href="#__codelineno-88-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-88-11" name="__codelineno-88-11" href="#__codelineno-88-11"></a><span class="w">    </span><span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-88-11" name="__codelineno-88-11" href="#__codelineno-88-11"></a><span class="w">    </span><span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-88-12" name="__codelineno-88-12" href="#__codelineno-88-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-13" name="__codelineno-88-13" href="#__codelineno-88-13"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-88-14" name="__codelineno-88-14" href="#__codelineno-88-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5730,7 +5730,7 @@
 <a id="__codelineno-89-9" name="__codelineno-89-9" href="#__codelineno-89-9"></a>    <span class="k">if</span> <span class="n">node</span> <span class="p">==</span> <span class="kc">nil</span> <span class="p">{</span>
 <a id="__codelineno-89-10" name="__codelineno-89-10" href="#__codelineno-89-10"></a>        <span class="k">return</span> <span class="kc">nil</span>
 <a id="__codelineno-89-11" name="__codelineno-89-11" href="#__codelineno-89-11"></a>    <span class="p">}</span>
-<a id="__codelineno-89-12" name="__codelineno-89-12" href="#__codelineno-89-12"></a>    <span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-89-12" name="__codelineno-89-12" href="#__codelineno-89-12"></a>    <span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-89-13" name="__codelineno-89-13" href="#__codelineno-89-13"></a>    <span class="k">if</span> <span class="n">val</span> <span class="o">&lt;</span> <span class="n">node</span><span class="p">!.</span><span class="n">val</span> <span class="p">{</span>
 <a id="__codelineno-89-14" name="__codelineno-89-14" href="#__codelineno-89-14"></a>        <span class="n">node</span><span class="p">?.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span> <span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-89-15" name="__codelineno-89-15" href="#__codelineno-89-15"></a>    <span class="p">}</span> <span class="k">else</span> <span class="k">if</span> <span class="n">val</span> <span class="o">&gt;</span> <span class="n">node</span><span class="p">!.</span><span class="n">val</span> <span class="p">{</span>
@@ -5773,7 +5773,7 @@
 <a id="__codelineno-90-6" name="__codelineno-90-6" href="#__codelineno-90-6"></a><span class="cm">/* 递归删除节点（辅助方法） */</span>
 <a id="__codelineno-90-7" name="__codelineno-90-7" href="#__codelineno-90-7"></a><span class="err">#</span><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-90-8" name="__codelineno-90-8" href="#__codelineno-90-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</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="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-90-9" name="__codelineno-90-9" href="#__codelineno-90-9"></a><span class="w">    </span><span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-90-9" name="__codelineno-90-9" href="#__codelineno-90-9"></a><span class="w">    </span><span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-90-10" name="__codelineno-90-10" href="#__codelineno-90-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-90-11" name="__codelineno-90-11" href="#__codelineno-90-11"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-90-12" name="__codelineno-90-12" href="#__codelineno-90-12"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
@@ -5811,7 +5811,7 @@
 <a id="__codelineno-91-6" name="__codelineno-91-6" href="#__codelineno-91-6"></a><span class="cm">/* 递归删除节点（辅助方法） */</span>
 <a id="__codelineno-91-7" name="__codelineno-91-7" href="#__codelineno-91-7"></a><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-91-8" name="__codelineno-91-8" href="#__codelineno-91-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</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="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-91-9" name="__codelineno-91-9" href="#__codelineno-91-9"></a><span class="w">    </span><span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-91-9" name="__codelineno-91-9" href="#__codelineno-91-9"></a><span class="w">    </span><span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-91-10" name="__codelineno-91-10" href="#__codelineno-91-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-91-11" name="__codelineno-91-11" href="#__codelineno-91-11"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-91-12" name="__codelineno-91-12" href="#__codelineno-91-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5853,7 +5853,7 @@
 <a id="__codelineno-92-6" name="__codelineno-92-6" href="#__codelineno-92-6"></a><span class="cm">/* 递归删除节点（辅助方法） */</span>
 <a id="__codelineno-92-7" name="__codelineno-92-7" href="#__codelineno-92-7"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-8" name="__codelineno-92-8" href="#__codelineno-92-8"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</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="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-92-9" name="__codelineno-92-9" href="#__codelineno-92-9"></a><span class="w">  </span><span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-92-9" name="__codelineno-92-9" href="#__codelineno-92-9"></a><span class="w">  </span><span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-92-10" name="__codelineno-92-10" href="#__codelineno-92-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-92-11" name="__codelineno-92-11" href="#__codelineno-92-11"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-92-12" name="__codelineno-92-12" href="#__codelineno-92-12"></a><span class="w">  </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
@@ -5895,7 +5895,7 @@
 <a id="__codelineno-93-7" name="__codelineno-93-7" href="#__codelineno-93-7"></a><span class="k">fn</span> <span class="nf">remove_helper</span><span class="p">(</span><span class="n">node</span>: <span class="nc">OptionTreeNodeRc</span><span class="p">,</span><span class="w"> </span><span class="n">val</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="nc">OptionTreeNodeRc</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-8" name="__codelineno-93-8" href="#__codelineno-93-8"></a><span class="w">    </span><span class="k">match</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-9" name="__codelineno-93-9" href="#__codelineno-93-9"></a><span class="w">        </span><span class="nb">Some</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-93-10" name="__codelineno-93-10" href="#__codelineno-93-10"></a><span class="w">            </span><span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-93-10" name="__codelineno-93-10" href="#__codelineno-93-10"></a><span class="w">            </span><span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-93-11" name="__codelineno-93-11" href="#__codelineno-93-11"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-12" name="__codelineno-93-12" href="#__codelineno-93-12"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-93-13" name="__codelineno-93-13" href="#__codelineno-93-13"></a><span class="w">                </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span>::<span class="n">remove_helper</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
@@ -5954,7 +5954,7 @@
 <a id="__codelineno-94-10" name="__codelineno-94-10" href="#__codelineno-94-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-11" name="__codelineno-94-11" href="#__codelineno-94-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-94-12" name="__codelineno-94-12" href="#__codelineno-94-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-94-13" name="__codelineno-94-13" href="#__codelineno-94-13"></a><span class="w">    </span><span class="cm">/* 1. 查找节点，并删除之 */</span>
+<a id="__codelineno-94-13" name="__codelineno-94-13" href="#__codelineno-94-13"></a><span class="w">    </span><span class="cm">/* 1. 查找节点并删除 */</span>
 <a id="__codelineno-94-14" name="__codelineno-94-14" href="#__codelineno-94-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-15" name="__codelineno-94-15" href="#__codelineno-94-15"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-94-16" name="__codelineno-94-16" href="#__codelineno-94-16"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6002,7 +6002,7 @@
 <a id="__codelineno-95-7" name="__codelineno-95-7" href="#__codelineno-95-7"></a><span class="k">fn</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">self</span><span class="o">:</span><span class="w"> </span><span class="o">*</span><span class="n">Self</span><span class="p">,</span><span class="w"> </span><span class="n">node_</span><span class="o">:</span><span class="w"> </span><span class="o">?*</span><span class="n">inc</span><span class="p">.</span><span class="n">TreeNode</span><span class="p">(</span><span class="n">T</span><span class="p">),</span><span class="w"> </span><span class="n">val</span><span class="o">:</span><span class="w"> </span><span class="n">T</span><span class="p">)</span><span class="w"> </span><span class="o">?*</span><span class="n">inc</span><span class="p">.</span><span class="n">TreeNode</span><span class="p">(</span><span class="n">T</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-8" name="__codelineno-95-8" href="#__codelineno-95-8"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node_</span><span class="p">;</span>
 <a id="__codelineno-95-9" name="__codelineno-95-9" href="#__codelineno-95-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</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="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-95-10" name="__codelineno-95-10" href="#__codelineno-95-10"></a><span class="w">    </span><span class="c1">// 1. 查找节点，并删除之</span>
+<a id="__codelineno-95-10" name="__codelineno-95-10" href="#__codelineno-95-10"></a><span class="w">    </span><span class="c1">// 1. 查找节点并删除</span>
 <a id="__codelineno-95-11" name="__codelineno-95-11" href="#__codelineno-95-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-12" name="__codelineno-95-12" href="#__codelineno-95-12"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="p">.</span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-95-13" name="__codelineno-95-13" href="#__codelineno-95-13"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -1830,7 +1830,7 @@
        
  
  <span class="md-ellipsis">
-    8.3 &nbsp; Top-K 问题
+    8.3 &nbsp; Top-k 问题
  </span>
  

@@ -3909,8 +3909,8 @@
 <h3 id="2">2. &nbsp; 插入节点<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
 <p>给定一个待插入元素 <code>num</code> ，为了保持二叉搜索树“左子树 &lt; 根节点 &lt; 右子树”的性质，插入操作流程如图 7-18 所示。</p>
 <ol>
-<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>
+<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 <code>None</code> ）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 <code>None</code> 的位置。</li>
 </ol>
 <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" class="animation-figure" 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>
@@ -3918,7 +3918,7 @@
 <p>在代码实现中，需要注意以下两点。</p>
 <ul>
 <li>二叉搜索树不允许存在重复节点，否则将违反其定义。因此，若待插入节点在树中已存在，则不执行插入，直接返回。</li>
-<li>为了实现插入节点，我们需要借助节点 <code>pre</code> 保存上一轮循环的节点。这样在遍历至 <span class="arithmatex">\(\text{None}\)</span> 时，我们可以获取到其父节点，从而完成节点插入操作。</li>
+<li>为了实现插入节点，我们需要借助节点 <code>pre</code> 保存上一轮循环的节点。这样在遍历至 <code>None</code> 时，我们可以获取到其父节点，从而完成节点插入操作。</li>
 </ul>
 <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>
 <div class="tabbed-content">
@@ -4310,9 +4310,7 @@
 </div>
 <p>与查找节点相同，插入节点使用 <span class="arithmatex">\(O(\log n)\)</span> 时间。</p>
 <h3 id="3">3. &nbsp; 删除节点<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
-<p>先在二叉树中查找到目标节点，再将其删除。</p>
-<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，我们根据目标节点的子节点数量，分 0、1 和 2 三种情况，执行对应的删除节点操作。</p>
+<p>先在二叉树中查找到目标节点，再将其删除。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。因此，我们根据目标节点的子节点数量，分 0、1 和 2 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><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 ）" class="animation-figure" 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>
@@ -1863,7 +1863,7 @@
        
  
  <span class="md-ellipsis">
-    8.3 &nbsp; Top-K 问题
+    8.3 &nbsp; Top-k 问题
  </span>
  

@@ -3853,7 +3853,7 @@
 <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 <span class="arithmatex">\(\text{None}\)</span> 。</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>
@@ -1827,7 +1827,7 @@
        
  
  <span class="md-ellipsis">
-    8.3 &nbsp; Top-K 问题
+    8.3 &nbsp; Top-k 问题
  </span>
  

@@ -3615,7 +3615,7 @@
 <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, BFS」，它体现了一种“一圈一圈向外扩展”的逐层遍历方式。</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><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>

@@ -3905,11 +3905,11 @@
 </div>
 <h3 id="2">2. &nbsp; 复杂度分析<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
 <ul>
-<li><strong>时间复杂度 <span class="arithmatex">\(O(n)\)</span></strong> ：所有节点被访问一次，使用 <span class="arithmatex">\(O(n)\)</span> 时间，其中 <span class="arithmatex">\(n\)</span> 为节点数量。</li>
-<li><strong>空间复杂度 <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>
+<li><strong>时间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> ：所有节点被访问一次，使用 <span class="arithmatex">\(O(n)\)</span> 时间，其中 <span class="arithmatex">\(n\)</span> 为节点数量。</li>
+<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>
 </ul>
 <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, 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「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>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>
@@ -4357,8 +4357,8 @@

 <h3 id="2_1">2. &nbsp; 复杂度分析<a class="headerlink" href="#2_1" title="Permanent link">&para;</a></h3>
 <ul>
-<li><strong>时间复杂度 <span class="arithmatex">\(O(n)\)</span></strong> ：所有节点被访问一次，使用 <span class="arithmatex">\(O(n)\)</span> 时间。</li>
-<li><strong>空间复杂度 <span class="arithmatex">\(O(n)\)</span></strong> ：在最差情况下，即树退化为链表时，递归深度达到 <span class="arithmatex">\(n\)</span> ，系统占用 <span class="arithmatex">\(O(n)\)</span> 栈帧空间。</li>
+<li><strong>时间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> ：所有节点被访问一次，使用 <span class="arithmatex">\(O(n)\)</span> 时间。</li>
+<li><strong>空间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> ：在最差情况下，即树退化为链表时，递归深度达到 <span class="arithmatex">\(n\)</span> ，系统占用 <span class="arithmatex">\(O(n)\)</span> 栈帧空间。</li>
 </ul>

 <!-- Source file information -->
@@ -1721,7 +1721,7 @@
        
  
  <span class="md-ellipsis">
-    8.3 &nbsp; Top-K 问题
+    8.3 &nbsp; Top-k 问题
  </span>
  

@@ -1779,7 +1779,7 @@
        
  
  <span class="md-ellipsis">
-    8.3 &nbsp; Top-K 问题
+    8.3 &nbsp; Top-k 问题
  </span>