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90
en/docs/chapter_tree/array_representation_of_tree.md
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@@ -237,7 +237,95 @@ The following code implements a binary tree based on array representation, inclu
=== "C++"
```cpp title="array_binary_tree.cpp"
[class]{ArrayBinaryTree}-[func]{}
/* Array-based binary tree class */
class ArrayBinaryTree {
public:
/* Constructor */
ArrayBinaryTree(vector<int> arr) {
tree = arr;
}
/* List capacity */
int size() {
return tree.size();
}
/* Get the value of the node at index i */
int val(int i) {
// If index is out of bounds, return INT_MAX, representing a null
if (i < 0 || i >= size())
return INT_MAX;
return tree[i];
}
/* Get the index of the left child of the node at index i */
int left(int i) {
return 2 * i + 1;
}
/* Get the index of the right child of the node at index i */
int right(int i) {
return 2 * i + 2;
}
/* Get the index of the parent of the node at index i */
int parent(int i) {
return (i - 1) / 2;
}
/* Level-order traversal */
vector<int> levelOrder() {
vector<int> res;
// Traverse array
for (int i = 0; i < size(); i++) {
if (val(i) != INT_MAX)
res.push_back(val(i));
}
return res;
}
/* Pre-order traversal */
vector<int> preOrder() {
vector<int> res;
dfs(0, "pre", res);
return res;
}
/* In-order traversal */
vector<int> inOrder() {
vector<int> res;
dfs(0, "in", res);
return res;
}
/* Post-order traversal */
vector<int> postOrder() {
vector<int> res;
dfs(0, "post", res);
return res;
}
private:
vector<int> tree;
/* Depth-first traversal */
void dfs(int i, string order, vector<int> &res) {
// If it is an empty spot, return
if (val(i) == INT_MAX)
return;
// Pre-order traversal
if (order == "pre")
res.push_back(val(i));
dfs(left(i), order, res);
// In-order traversal
if (order == "in")
res.push_back(val(i));
dfs(right(i), order, res);
// Post-order traversal
if (order == "post")
res.push_back(val(i));
}
};
```
=== "Java"

147
en/docs/chapter_tree/avl_tree.md
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@@ -252,9 +252,17 @@ The "node height" refers to the distance from that node to its farthest leaf nod
=== "C++"
```cpp title="avl_tree.cpp"
[class]{AVLTree}-[func]{height}
/* Get node height */
int height(TreeNode *node) {
// Empty node height is -1, leaf node height is 0
return node == nullptr ? -1 : node->height;
}
[class]{AVLTree}-[func]{updateHeight}
/* Update node height */
void updateHeight(TreeNode *node) {
// Node height equals the height of the tallest subtree + 1
node->height = max(height(node->left), height(node->right)) + 1;
}
```
=== "Java"
@@ -380,7 +388,14 @@ The <u>balance factor</u> of a node is defined as the height of the node's left
=== "C++"
```cpp title="avl_tree.cpp"
[class]{AVLTree}-[func]{balanceFactor}
/* Get balance factor */
int balanceFactor(TreeNode *node) {
// Empty node balance factor is 0
if (node == nullptr)
return 0;
// Node balance factor = left subtree height - right subtree height
return height(node->left) - height(node->right);
}
```
=== "Java"
@@ -518,7 +533,19 @@ As shown in Figure 7-27, when the `child` node has a right child (denoted as `gr
=== "C++"
```cpp title="avl_tree.cpp"
[class]{AVLTree}-[func]{rightRotate}
/* Right rotation operation */
TreeNode *rightRotate(TreeNode *node) {
TreeNode *child = node->left;
TreeNode *grandChild = child->right;
// Rotate node to the right around child
child->right = node;
node->left = grandChild;
// Update node height
updateHeight(node);
updateHeight(child);
// Return the root of the subtree after rotation
return child;
}
```
=== "Java"
@@ -641,7 +668,19 @@ It can be observed that **the right and left rotation operations are logically s
=== "C++"
```cpp title="avl_tree.cpp"
[class]{AVLTree}-[func]{leftRotate}
/* Left rotation operation */
TreeNode *leftRotate(TreeNode *node) {
TreeNode *child = node->right;
TreeNode *grandChild = child->left;
// Rotate node to the left around child
child->left = node;
node->right = grandChild;
// Update node height
updateHeight(node);
updateHeight(child);
// Return the root of the subtree after rotation
return child;
}
```
=== "Java"
@@ -801,7 +840,35 @@ For convenience, we encapsulate the rotation operations into a function. **With
=== "C++"
```cpp title="avl_tree.cpp"
[class]{AVLTree}-[func]{rotate}
/* Perform rotation operation to restore balance to the subtree */
TreeNode *rotate(TreeNode *node) {
// Get the balance factor of node
int _balanceFactor = balanceFactor(node);
// Left-leaning tree
if (_balanceFactor > 1) {
if (balanceFactor(node->left) >= 0) {
// Right rotation
return rightRotate(node);
} else {
// First left rotation then right rotation
node->left = leftRotate(node->left);
return rightRotate(node);
}
}
// Right-leaning tree
if (_balanceFactor < -1) {
if (balanceFactor(node->right) <= 0) {
// Left rotation
return leftRotate(node);
} else {
// First right rotation then left rotation
node->right = rightRotate(node->right);
return leftRotate(node);
}
}
// Balanced tree, no rotation needed, return
return node;
}
```
=== "Java"
@@ -938,9 +1005,28 @@ The node insertion operation in AVL trees is similar to that in binary search tr
=== "C++"
```cpp title="avl_tree.cpp"
[class]{AVLTree}-[func]{insert}
/* Insert node */
void insert(int val) {
root = insertHelper(root, val);
}
[class]{AVLTree}-[func]{insertHelper}
/* Recursively insert node (helper method) */
TreeNode *insertHelper(TreeNode *node, int val) {
if (node == nullptr)
return new TreeNode(val);
/* 1. Find insertion position and insert node */
if (val < node->val)
node->left = insertHelper(node->left, val);
else if (val > node->val)
node->right = insertHelper(node->right, val);
else
return node; // Do not insert duplicate nodes, return
updateHeight(node); // Update node height
/* 2. Perform rotation operation to restore balance to the subtree */
node = rotate(node);
// Return the root node of the subtree
return node;
}
```
=== "Java"
@@ -1103,9 +1189,50 @@ Similarly, based on the method of removing nodes in binary search trees, rotatio
=== "C++"
```cpp title="avl_tree.cpp"
[class]{AVLTree}-[func]{remove}
/* Remove node */
void remove(int val) {
root = removeHelper(root, val);
}
[class]{AVLTree}-[func]{removeHelper}
/* Recursively remove node (helper method) */
TreeNode *removeHelper(TreeNode *node, int val) {
if (node == nullptr)
return nullptr;
/* 1. Find and remove the node */
if (val < node->val)
node->left = removeHelper(node->left, val);
else if (val > node->val)
node->right = removeHelper(node->right, val);
else {
if (node->left == nullptr || node->right == nullptr) {
TreeNode *child = node->left != nullptr ? node->left : node->right;
// Number of child nodes = 0, remove node and return
if (child == nullptr) {
delete node;
return nullptr;
}
// Number of child nodes = 1, remove node
else {
delete node;
node = child;
}
} else {
// Number of child nodes = 2, remove the next node in in-order traversal and replace the current node with it
TreeNode *temp = node->right;
while (temp->left != nullptr) {
temp = temp->left;
}
int tempVal = temp->val;
node->right = removeHelper(node->right, temp->val);
node->val = tempVal;
}
}
updateHeight(node); // Update node height
/* 2. Perform rotation operation to restore balance to the subtree */
node = rotate(node);
// Return the root node of the subtree
return node;
}
```
=== "Java"

102
en/docs/chapter_tree/binary_search_tree.md
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@@ -64,7 +64,24 @@ The search operation in a binary search tree works on the same principle as the
=== "C++"
```cpp title="binary_search_tree.cpp"
[class]{BinarySearchTree}-[func]{search}
/* Search node */
TreeNode *search(int num) {
TreeNode *cur = root;
// Loop find, break after passing leaf nodes
while (cur != nullptr) {
// Target node is in cur's right subtree
if (cur->val < num)
cur = cur->right;
// Target node is in cur's left subtree
else if (cur->val > num)
cur = cur->left;
// Found target node, break loop
else
break;
}
// Return target node
return cur;
}
```
=== "Java"
@@ -205,7 +222,34 @@ In the code implementation, note the following two points.
=== "C++"
```cpp title="binary_search_tree.cpp"
[class]{BinarySearchTree}-[func]{insert}
/* Insert node */
void insert(int num) {
// If tree is empty, initialize root node
if (root == nullptr) {
root = new TreeNode(num);
return;
}
TreeNode *cur = root, *pre = nullptr;
// Loop find, break after passing leaf nodes
while (cur != nullptr) {
// Found duplicate node, thus return
if (cur->val == num)
return;
pre = cur;
// Insertion position is in cur's right subtree
if (cur->val < num)
cur = cur->right;
// Insertion position is in cur's left subtree
else
cur = cur->left;
}
// Insert node
TreeNode *node = new TreeNode(num);
if (pre->val < num)
pre->right = node;
else
pre->left = node;
}
```
=== "Java"
@@ -401,7 +445,59 @@ The operation of removing a node also uses $O(\log n)$ time, where finding the n
=== "C++"
```cpp title="binary_search_tree.cpp"
[class]{BinarySearchTree}-[func]{remove}
/* Remove node */
void remove(int num) {
// If tree is empty, return
if (root == nullptr)
return;
TreeNode *cur = root, *pre = nullptr;
// Loop find, break after passing leaf nodes
while (cur != nullptr) {
// Found node to be removed, break loop
if (cur->val == num)
break;
pre = cur;
// Node to be removed is in cur's right subtree
if (cur->val < num)
cur = cur->right;
// Node to be removed is in cur's left subtree
else
cur = cur->left;
}
// If no node to be removed, return
if (cur == nullptr)
return;
// Number of child nodes = 0 or 1
if (cur->left == nullptr || cur->right == nullptr) {
// When the number of child nodes = 0 / 1, child = nullptr / that child node
TreeNode *child = cur->left != nullptr ? cur->left : cur->right;
// Remove node cur
if (cur != root) {
if (pre->left == cur)
pre->left = child;
else
pre->right = child;
} else {
// If the removed node is the root, reassign the root
root = child;
}
// Free memory
delete cur;
}
// Number of child nodes = 2
else {
// Get the next node in in-order traversal of cur
TreeNode *tmp = cur->right;
while (tmp->left != nullptr) {
tmp = tmp->left;
}
int tmpVal = tmp->val;
// Recursively remove node tmp
remove(tmp->val);
// Replace cur with tmp
cur->val = tmpVal;
}
}
```
=== "Java"

49
en/docs/chapter_tree/binary_tree_traversal.md
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@@ -45,7 +45,24 @@ Breadth-first traversal is usually implemented with the help of a "queue". The q
=== "C++"
```cpp title="binary_tree_bfs.cpp"
[class]{}-[func]{levelOrder}
/* Level-order traversal */
vector<int> levelOrder(TreeNode *root) {
// Initialize queue, add root node
queue<TreeNode *> queue;
queue.push(root);
// Initialize a list to store the traversal sequence
vector<int> vec;
while (!queue.empty()) {
TreeNode *node = queue.front();
queue.pop(); // Queue dequeues
vec.push_back(node->val); // Save node value
if (node->left != nullptr)
queue.push(node->left); // Left child node enqueues
if (node->right != nullptr)
queue.push(node->right); // Right child node enqueues
}
return vec;
}
```
=== "Java"
@@ -189,11 +206,35 @@ Depth-first search is usually implemented based on recursion:
=== "C++"
```cpp title="binary_tree_dfs.cpp"
[class]{}-[func]{preOrder}
/* Pre-order traversal */
void preOrder(TreeNode *root) {
if (root == nullptr)
return;
// Visit priority: root node -> left subtree -> right subtree
vec.push_back(root->val);
preOrder(root->left);
preOrder(root->right);
}
[class]{}-[func]{inOrder}
/* In-order traversal */
void inOrder(TreeNode *root) {
if (root == nullptr)
return;
// Visit priority: left subtree -> root node -> right subtree
inOrder(root->left);
vec.push_back(root->val);
inOrder(root->right);
}
[class]{}-[func]{postOrder}
/* Post-order traversal */
void postOrder(TreeNode *root) {
if (root == nullptr)
return;
// Visit priority: left subtree -> right subtree -> root node
postOrder(root->left);
postOrder(root->right);
vec.push_back(root->val);
}
```
=== "Java"