build

en/docs/chapter_tree/avl_tree.md

@@ -18,11 +18,11 @@ For example, in the perfect binary tree shown in Figure 7-25, after inserting tw
 
 <p align="center"> Figure 7-25 &nbsp; Degradation of an AVL tree after inserting nodes </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 $O(\log n)$ 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.
+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 $O(\log n)$ 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.
 
 ## 7.5.1 &nbsp; Common terminology in AVL trees
 
-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".
+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>.
 
 ### 1. &nbsp; Node height
 
@@ -489,7 +489,7 @@ The "node height" refers to the distance from that node to its farthest leaf nod
 
 ### 2. &nbsp; Node balance factor
 
-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 $0$. We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:
+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 $0$. We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:
 
 === "Python"
 

en/docs/chapter_tree/binary_search_tree.md

@@ -4,9 +4,9 @@ comments: true
 
 # 7.4   Binary search tree
 
-As shown in Figure 7-16, a "binary search tree" satisfies the following conditions.
+As shown in Figure 7-16, a <u>binary search tree</u> satisfies the following conditions.
 
-1. For the root node, the value of all nodes in the left subtree < the value of the root node < the value of all nodes in the right subtree.
+1. For the root node, the value of all nodes in the left subtree $<$ the value of the root node $<$ the value of all nodes in the right subtree.
 2. The left and right subtrees of any node are also binary search trees, i.e., they satisfy condition `1.` as well.
 
 ![Binary search tree](binary_search_tree.assets/binary_search_tree.png){ class="animation-figure" }
@@ -885,7 +885,7 @@ As shown in Figure 7-20, when the degree of the node to be removed is $1$, repla
 
 <p align="center"> Figure 7-20 &nbsp; Removing a node in a binary search tree (degree 1) </p>
 
-When the degree of the node to be removed is $2$, we cannot remove it directly, but need to use a node to replace it. To maintain the property of the binary search tree "left subtree < root node < right subtree," **this node can be either the smallest node of the right subtree or the largest node of the left subtree**.
+When the degree of the node to be removed is $2$, we cannot remove it directly, but need to use a node to replace it. To maintain the property of the binary search tree "left subtree $<$ root node $<$ right subtree," **this node can be either the smallest node of the right subtree or the largest node of the left subtree**.
 
 Assuming we choose the smallest node of the right subtree (the next node in in-order traversal), then the removal operation proceeds as shown in Figure 7-21.
 
@@ -1705,7 +1705,7 @@ The operation of removing a node also uses $O(\log n)$ time, where finding the n
 
 ### 4. &nbsp; In-order traversal is ordered
 
-As shown in Figure 7-22, the in-order traversal of a binary tree follows the "left $\rightarrow$ root $\rightarrow$ right" traversal order, and a binary search tree satisfies the size relationship "left child node < root node < right child node".
+As shown in Figure 7-22, the in-order traversal of a binary tree follows the "left $\rightarrow$ root $\rightarrow$ right" traversal order, and a binary search tree satisfies the size relationship "left child node $<$ root node $<$ right child node".
 
 This means that in-order traversal in a binary search tree always traverses the next smallest node first, thus deriving an important property: **The in-order traversal sequence of a binary search tree is ascending**.
 

en/docs/chapter_tree/binary_tree.md

@@ -4,7 +4,7 @@ comments: true
 
 # 7.1   Binary tree
 
-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.
+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.
 
 === "Python"
 
@@ -202,7 +202,7 @@ A "binary tree" is a non-linear data structure that represents the ancestral and
 
     ```
 
-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.
+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.
 
 **In a binary tree, except for leaf nodes, all other nodes contain child nodes and non-empty subtrees.** 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."
 
@@ -214,14 +214,14 @@ Each node has two references (pointers), pointing to the "left-child node" and "
 
 The commonly used terminology of binary trees is shown in Figure 7-2.
 
-- "Root node": The node at the top level of the binary tree, which has no parent node.
-- "Leaf node": A node with no children, both of its pointers point to `None`.
-- "Edge": The line segment connecting two nodes, i.e., node reference (pointer).
-- The "level" of a node: Incrementing from top to bottom, with the root node's level being 1.
-- The "degree" of a node: The number of a node's children. In a binary tree, the degree can be 0, 1, or 2.
-- The "height" of a binary tree: The number of edges passed from the root node to the farthest leaf node.
-- The "depth" of a node: The number of edges passed from the root node to the node.
-- The "height" of a node: The number of edges from the farthest leaf node to the node.
+- <u>Root node</u>: The node at the top level of the binary tree, which has no parent node.
+- <u>Leaf node</u>: A node with no children, both of its pointers point to `None`.
+- <u>Edge</u>: The line segment connecting two nodes, i.e., node reference (pointer).
+- The <u>level</u> of a node: Incrementing from top to bottom, with the root node's level being 1.
+- 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.
+- The <u>height</u> of a binary tree: The number of edges passed from the root node to the farthest leaf node.
+- The <u>depth</u> of a node: The number of edges passed from the root node to the node.
+- The <u>height</u> of a node: The number of edges from the farthest leaf node to the node.
 
 ![Common Terminology of Binary Trees](binary_tree.assets/binary_tree_terminology.png){ class="animation-figure" }
 
@@ -625,11 +625,11 @@ Similar to a linked list, inserting and removing nodes in a binary tree can be a
 
 ### 1. &nbsp; Perfect binary tree
 
-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 $0$, and the degree of all other nodes is $2$; if the tree's height is $h$, then the total number of nodes is $2^{h+1} - 1$, showing a standard exponential relationship, reflecting the common phenomenon of cell division in nature.
+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 $0$, and the degree of all other nodes is $2$; if the tree's height is $h$, then the total number of nodes is $2^{h+1} - 1$, showing a standard exponential relationship, reflecting the common phenomenon of cell division in nature.
 
 !!! tip
 
-    Please note that in the Chinese community, a perfect binary tree is often referred to as a "full binary tree."
+    Please note that in the Chinese community, a perfect binary tree is often referred to as a <u>full binary tree</u>.
 
 ![Perfect binary tree](binary_tree.assets/perfect_binary_tree.png){ class="animation-figure" }
 
@@ -637,7 +637,7 @@ As shown in Figure 7-4, in a "perfect binary tree," all levels of nodes are full
 
 ### 2. &nbsp; Complete binary tree
 
-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.
+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.
 
 ![Complete binary tree](binary_tree.assets/complete_binary_tree.png){ class="animation-figure" }
 
@@ -645,7 +645,7 @@ As shown in Figure 7-5, a "complete binary tree" has only the bottom level nodes
 
 ### 3. &nbsp; Full binary tree
 
-As shown in Figure 7-6, a "full binary tree" has all nodes except leaf nodes having two children.
+As shown in Figure 7-6, a <u>full binary tree</u> has all nodes except leaf nodes having two children.
 
 ![Full binary tree](binary_tree.assets/full_binary_tree.png){ class="animation-figure" }
 
@@ -653,7 +653,7 @@ As shown in Figure 7-6, a "full binary tree" has all nodes except leaf nodes hav
 
 ### 4. &nbsp; Balanced binary tree
 
-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.
+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.
 
 ![Balanced binary tree](binary_tree.assets/balanced_binary_tree.png){ class="animation-figure" }
 

en/docs/chapter_tree/binary_tree_traversal.md

@@ -10,9 +10,9 @@ Common traversal methods for binary trees include level-order traversal, preorde
 
 ## 7.2.1   Level-order traversal
 
-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.
+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.
 
-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.
+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.
 
 ![Level-order traversal of a binary tree](binary_tree_traversal.assets/binary_tree_bfs.png){ class="animation-figure" }
 
@@ -378,7 +378,7 @@ Breadth-first traversal is usually implemented with the help of a "queue". The q
 
 ## 7.2.2 &nbsp; Preorder, inorder, and postorder traversal
 
-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.
+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.
 
 Figure 7-10 shows the working principle of performing a depth-first traversal on a binary tree. **Depth-first traversal is like walking around the perimeter of the entire binary tree**, encountering three positions at each node, corresponding to preorder traversal, inorder traversal, and postorder traversal.
 

en/docs/chapter_tree/summary.md

@@ -30,7 +30,7 @@ Taking the binary search tree as an example, the operation of removing a node ne
 
 **Q**: Why are there three sequences: pre-order, in-order, and post-order for DFS traversal of a binary tree, and what are their uses?
 
-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 `left child node value < root node value < right child node value`, we can obtain an ordered node sequence by traversing the tree in the "left → root → right" priority.
+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 `left child node value < root node value < right child node value`, we can obtain an ordered node sequence by traversing the tree in the "left $\rightarrow$ root $\rightarrow$ right" priority.
 
 **Q**: In a right rotation operation that deals with the relationship between the imbalance nodes `node`, `child`, `grand_child`, isn't the connection between `node` and its parent node and the original link of `node` lost after the right rotation?