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      <div class="md-copyright">
  
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <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="Degradation of an AVL tree after inserting nodes" class="animation-figure" src="../avl_tree.assets/avltree_degradation_from_inserting_node.png" /></a></p>
 <p align="center"> Figure 7-25 &nbsp; Degradation of an AVL tree after inserting nodes </p>

-<p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the "AVL Tree" in their paper "An algorithm for the organization of information". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at <span class="arithmatex">\(O(\log n)\)</span> level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.</p>
+<p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the <u>AVL Tree</u> in their paper "An algorithm for the organization of information". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at <span class="arithmatex">\(O(\log n)\)</span> level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.</p>
 <h2 id="751-common-terminology-in-avl-trees">7.5.1 &nbsp; Common terminology in AVL trees<a class="headerlink" href="#751-common-terminology-in-avl-trees" title="Permanent link">&para;</a></h2>
-<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a "balanced binary search tree".</p>
+<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a <u>balanced binary search tree</u>.</p>
 <h3 id="1-node-height">1. &nbsp; Node height<a class="headerlink" href="#1-node-height" title="Permanent link">&para;</a></h3>
 <p>Since the operations related to AVL trees require obtaining node heights, we need to add a <code>height</code> variable to the node class:</p>
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 <h3 id="2-node-balance-factor">2. &nbsp; Node balance factor<a class="headerlink" href="#2-node-balance-factor" title="Permanent link">&para;</a></h3>
-<p>The "balance factor" of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as <span class="arithmatex">\(0\)</span>. We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:</p>
+<p>The <u>balance factor</u> of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as <span class="arithmatex">\(0\)</span>. We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:</p>
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@@ -6634,7 +6634,7 @@ aria-label="Footer"
      <div class="md-copyright">
  
    <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <!-- Page content -->
 <h1 id="71-binary-tree">7.1 &nbsp; Binary tree<a class="headerlink" href="#71-binary-tree" title="Permanent link">&para;</a></h1>
-<p>A "binary tree" is a non-linear data structure that represents the ancestral and descendent relationships, embodying the "divide and conquer" logic. Similar to a linked list, the basic unit of a binary tree is a node, each containing a value, a reference to the left child node, and a reference to the right child node.</p>
+<p>A <u>binary tree</u> is a non-linear data structure that represents the ancestral and descendent relationships, embodying the "divide and conquer" logic. Similar to a linked list, the basic unit of a binary tree is a node, each containing a value, a reference to the left child node, and a reference to the right child node.</p>
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-<p>Each node has two references (pointers), pointing to the "left-child node" and "right-child node," respectively. This node is called the "parent node" of these two child nodes. When given a node of a binary tree, we call the tree formed by this node's left child and all nodes under it the "left subtree" of this node. Similarly, the "right subtree" can be defined.</p>
+<p>Each node has two references (pointers), pointing to the <u>left-child node</u> and <u>right-child node</u>, respectively. This node is called the <u>parent node</u> of these two child nodes. When given a node of a binary tree, we call the tree formed by this node's left child and all nodes under it the <u>left subtree</u> of this node. Similarly, the <u>right subtree</u> can be defined.</p>
 <p><strong>In a binary tree, except for leaf nodes, all other nodes contain child nodes and non-empty subtrees.</strong> As shown in Figure 7-1, if "Node 2" is considered as the parent node, then its left and right child nodes are "Node 4" and "Node 5," respectively. The left subtree is "the tree formed by Node 4 and all nodes under it," and the right subtree is "the tree formed by Node 5 and all nodes under it."</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="Parent Node, child Node, subtree" class="animation-figure" src="../binary_tree.assets/binary_tree_definition.png" /></a></p>
 <p align="center"> Figure 7-1 &nbsp; Parent Node, child Node, subtree </p>
@@ -3940,14 +3940,14 @@
 <h2 id="711-common-terminology-of-binary-trees">7.1.1 &nbsp; Common terminology of binary trees<a class="headerlink" href="#711-common-terminology-of-binary-trees" title="Permanent link">&para;</a></h2>
 <p>The commonly used terminology of binary trees is shown in Figure 7-2.</p>
 <ul>
-<li>"Root node": The node at the top level of the binary tree, which has no parent node.</li>
-<li>"Leaf node": A node with no children, both of its pointers point to <code>None</code>.</li>
-<li>"Edge": The line segment connecting two nodes, i.e., node reference (pointer).</li>
-<li>The "level" of a node: Incrementing from top to bottom, with the root node's level being 1.</li>
-<li>The "degree" of a node: The number of a node's children. In a binary tree, the degree can be 0, 1, or 2.</li>
-<li>The "height" of a binary tree: The number of edges passed from the root node to the farthest leaf node.</li>
-<li>The "depth" of a node: The number of edges passed from the root node to the node.</li>
-<li>The "height" of a node: The number of edges from the farthest leaf node to the node.</li>
+<li><u>Root node</u>: The node at the top level of the binary tree, which has no parent node.</li>
+<li><u>Leaf node</u>: A node with no children, both of its pointers point to <code>None</code>.</li>
+<li><u>Edge</u>: The line segment connecting two nodes, i.e., node reference (pointer).</li>
+<li>The <u>level</u> of a node: Incrementing from top to bottom, with the root node's level being 1.</li>
+<li>The <u>degree</u> of a node: The number of a node's children. In a binary tree, the degree can be 0, 1, or 2.</li>
+<li>The <u>height</u> of a binary tree: The number of edges passed from the root node to the farthest leaf node.</li>
+<li>The <u>depth</u> of a node: The number of edges passed from the root node to the node.</li>
+<li>The <u>height</u> of a node: The number of edges from the farthest leaf node to the node.</li>
 </ul>
 <p><a class="glightbox" href="../binary_tree.assets/binary_tree_terminology.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Common Terminology of Binary Trees" class="animation-figure" src="../binary_tree.assets/binary_tree_terminology.png" /></a></p>
 <p align="center"> Figure 7-2 &nbsp; Common Terminology of Binary Trees </p>
@@ -4294,26 +4294,26 @@
 </div>
 <h2 id="713-common-types-of-binary-trees">7.1.3 &nbsp; Common types of binary trees<a class="headerlink" href="#713-common-types-of-binary-trees" title="Permanent link">&para;</a></h2>
 <h3 id="1-perfect-binary-tree">1. &nbsp; Perfect binary tree<a class="headerlink" href="#1-perfect-binary-tree" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-4, in a "perfect binary tree," all levels of nodes are fully filled. In a perfect binary tree, the degree of leaf nodes is <span class="arithmatex">\(0\)</span>, and the degree of all other nodes is <span class="arithmatex">\(2\)</span>; if the tree's height is <span class="arithmatex">\(h\)</span>, then the total number of nodes is <span class="arithmatex">\(2^{h+1} - 1\)</span>, showing a standard exponential relationship, reflecting the common phenomenon of cell division in nature.</p>
+<p>As shown in Figure 7-4, in a <u>perfect binary tree</u>, all levels of nodes are fully filled. In a perfect binary tree, the degree of leaf nodes is <span class="arithmatex">\(0\)</span>, and the degree of all other nodes is <span class="arithmatex">\(2\)</span>; if the tree's height is <span class="arithmatex">\(h\)</span>, then the total number of nodes is <span class="arithmatex">\(2^{h+1} - 1\)</span>, showing a standard exponential relationship, reflecting the common phenomenon of cell division in nature.</p>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Please note that in the Chinese community, a perfect binary tree is often referred to as a "full binary tree."</p>
+<p>Please note that in the Chinese community, a perfect binary tree is often referred to as a <u>full binary tree</u>.</p>
 </div>
 <p><a class="glightbox" href="../binary_tree.assets/perfect_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Perfect binary tree" class="animation-figure" src="../binary_tree.assets/perfect_binary_tree.png" /></a></p>
 <p align="center"> Figure 7-4 &nbsp; Perfect binary tree </p>

 <h3 id="2-complete-binary-tree">2. &nbsp; Complete binary tree<a class="headerlink" href="#2-complete-binary-tree" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-5, a "complete binary tree" has only the bottom level nodes not fully filled, and the bottom level nodes are filled as far left as possible.</p>
+<p>As shown in Figure 7-5, a <u>complete binary tree</u> has only the bottom level nodes not fully filled, and the bottom level nodes are filled as far left as possible.</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="Complete binary tree" class="animation-figure" src="../binary_tree.assets/complete_binary_tree.png" /></a></p>
 <p align="center"> Figure 7-5 &nbsp; Complete binary tree </p>

 <h3 id="3-full-binary-tree">3. &nbsp; Full binary tree<a class="headerlink" href="#3-full-binary-tree" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-6, a "full binary tree" has all nodes except leaf nodes having two children.</p>
+<p>As shown in Figure 7-6, a <u>full binary tree</u> has all nodes except leaf nodes having two children.</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="Full binary tree" class="animation-figure" src="../binary_tree.assets/full_binary_tree.png" /></a></p>
 <p align="center"> Figure 7-6 &nbsp; Full binary tree </p>

 <h3 id="4-balanced-binary-tree">4. &nbsp; Balanced binary tree<a class="headerlink" href="#4-balanced-binary-tree" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-7, in a "balanced binary tree," the absolute difference in height between the left and right subtrees of any node does not exceed 1.</p>
+<p>As shown in Figure 7-7, in a <u>balanced binary tree</u>, the absolute difference in height between the left and right subtrees of any node does not exceed 1.</p>
 <p><a class="glightbox" href="../binary_tree.assets/balanced_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Balanced binary tree" class="animation-figure" src="../binary_tree.assets/balanced_binary_tree.png" /></a></p>
 <p align="center"> Figure 7-7 &nbsp; Balanced binary tree </p>

@@ -4549,7 +4549,7 @@ aria-label="Footer"
      <div class="md-copyright">
  
    <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
    </div>
  
  
@@ -3690,8 +3690,8 @@
 <p>From the perspective of physical structure, a tree is a data structure based on linked lists, hence its traversal method involves accessing nodes one by one through pointers. However, a tree is a non-linear data structure, which makes traversing a tree more complex than traversing a linked list, requiring the assistance of search algorithms to achieve.</p>
 <p>Common traversal methods for binary trees include level-order traversal, preorder traversal, inorder traversal, and postorder traversal, among others.</p>
 <h2 id="721-level-order-traversal">7.2.1 &nbsp; Level-order traversal<a class="headerlink" href="#721-level-order-traversal" title="Permanent link">&para;</a></h2>
-<p>As shown in Figure 7-9, "level-order traversal" traverses the binary tree from top to bottom, layer by layer, and accesses nodes in each layer in a left-to-right order.</p>
-<p>Level-order traversal essentially belongs to "breadth-first traversal", also known as "breadth-first search (BFS)", which embodies a "circumferentially outward expanding" layer-by-layer traversal method.</p>
+<p>As shown in Figure 7-9, <u>level-order traversal</u> traverses the binary tree from top to bottom, layer by layer, and accesses nodes in each layer in a left-to-right order.</p>
+<p>Level-order traversal essentially belongs to <u>breadth-first traversal</u>, also known as <u>breadth-first search (BFS)</u>, which embodies a "circumferentially outward expanding" layer-by-layer traversal method.</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="Level-order traversal of a binary tree" class="animation-figure" src="../binary_tree_traversal.assets/binary_tree_bfs.png" /></a></p>
 <p align="center"> Figure 7-9 &nbsp; Level-order traversal of a binary tree </p>

@@ -4028,7 +4028,7 @@
 <li><strong>Space complexity is <span class="arithmatex">\(O(n)\)</span></strong>: In the worst case, i.e., a full binary tree, before traversing to the lowest level, the queue can contain at most <span class="arithmatex">\((n + 1) / 2\)</span> nodes at the same time, occupying <span class="arithmatex">\(O(n)\)</span> space.</li>
 </ul>
 <h2 id="722-preorder-inorder-and-postorder-traversal">7.2.2 &nbsp; Preorder, inorder, and postorder traversal<a class="headerlink" href="#722-preorder-inorder-and-postorder-traversal" title="Permanent link">&para;</a></h2>
-<p>Correspondingly, preorder, inorder, and postorder traversal all belong to "depth-first traversal", also known as "depth-first search (DFS)", which embodies a "proceed to the end first, then backtrack and continue" traversal method.</p>
+<p>Correspondingly, preorder, inorder, and postorder traversal all belong to <u>depth-first traversal</u>, also known as <u>depth-first search (DFS)</u>, which embodies a "proceed to the end first, then backtrack and continue" traversal method.</p>
 <p>Figure 7-10 shows the working principle of performing a depth-first traversal on a binary tree. <strong>Depth-first traversal is like walking around the perimeter of the entire binary tree</strong>, encountering three positions at each node, corresponding to preorder traversal, inorder traversal, and postorder traversal.</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="Preorder, inorder, and postorder traversal of a binary search tree" class="animation-figure" src="../binary_tree_traversal.assets/binary_tree_dfs.png" /></a></p>
 <p align="center"> Figure 7-10 &nbsp; Preorder, inorder, and postorder traversal of a binary search tree </p>
@@ -4732,7 +4732,7 @@ aria-label="Footer"
      <div class="md-copyright">
  
    <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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      <div class="md-copyright">
  
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-      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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 <p><strong>Q</strong>: The insertion and removal in a binary tree are generally completed by a set of operations. What does "a set of operations" refer to here? Can it be understood as the release of resources of the child nodes?</p>
 <p>Taking the binary search tree as an example, the operation of removing a node needs to be handled in three different scenarios, each requiring multiple steps of node operations.</p>
 <p><strong>Q</strong>: Why are there three sequences: pre-order, in-order, and post-order for DFS traversal of a binary tree, and what are their uses?</p>
-<p>Similar to sequential and reverse traversal of arrays, pre-order, in-order, and post-order traversals are three methods of traversing a binary tree, allowing us to obtain a traversal result in a specific order. For example, in a binary search tree, since the node sizes satisfy <code>left child node value &lt; root node value &lt; right child node value</code>, we can obtain an ordered node sequence by traversing the tree in the "left → root → right" priority.</p>
+<p>Similar to sequential and reverse traversal of arrays, pre-order, in-order, and post-order traversals are three methods of traversing a binary tree, allowing us to obtain a traversal result in a specific order. For example, in a binary search tree, since the node sizes satisfy <code>left child node value &lt; root node value &lt; right child node value</code>, we can obtain an ordered node sequence by traversing the tree in the "left <span class="arithmatex">\(\rightarrow\)</span> root <span class="arithmatex">\(\rightarrow\)</span> right" priority.</p>
 <p><strong>Q</strong>: In a right rotation operation that deals with the relationship between the imbalance nodes <code>node</code>, <code>child</code>, <code>grand_child</code>, isn't the connection between <code>node</code> and its parent node and the original link of <code>node</code> lost after the right rotation?</p>
 <p>We need to view this problem from a recursive perspective. The <code>right_rotate(root)</code> operation passes the root node of the subtree and eventually returns the root node of the rotated subtree with <code>return child</code>. The connection between the subtree's root node and its parent node is established after this function returns, which is outside the scope of the right rotation operation's maintenance.</p>
 <p><strong>Q</strong>: In C++, functions are divided into <code>private</code> and <code>public</code> sections. What considerations are there for this? Why are the <code>height()</code> function and the <code>updateHeight()</code> function placed in <code>public</code> and <code>private</code>, respectively?</p>
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