add binary_tree and avl_tree python code

codes/python/chapter_tree/avl_tree.py

@@ -0,0 +1,281 @@
+import sys, os.path as osp
+import typing
+
+sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
+from include import *
+
+
+class AVLTreeNode:
+    def __init__(
+        self,
+        val=None,
+        height: int = 0,
+        left: typing.Optional["AVLTreeNode"] = None,
+        right: typing.Optional["AVLTreeNode"] = None,
+    ):
+        self.val = val
+        self.height = height
+        self.left = left
+        self.right = right
+
+    def __str__(self):
+        val = self.val
+        left_val = self.left.val if self.left else None
+        right_val = self.right.val if self.right else None
+        return "<AVLTreeNode: {}, leftAVLTreeNode: {}, rightAVLTreeNode: {}>".format(
+            val, left_val, right_val
+        )
+
+
+class AVLTree:
+    def __init__(self, root: typing.Optional[AVLTreeNode] = None):
+        self.root = root
+
+    @staticmethod
+    def height(node: typing.Optional[AVLTreeNode]) -> int:
+        """
+        获取结点高度
+        Args:
+            node:起始结点
+
+        Returns: 高度 or -1
+
+        """
+        # 空结点高度为 -1 ，叶结点高度为 0
+        if node is not None:
+            return node.height
+        return -1
+
+    def __update_height(self, node: AVLTreeNode):
+        """
+        更新结点高度
+        Args:
+            node: 要更新高度的结点
+
+        Returns: None
+
+        """
+        # 结点高度等于最高子树高度 + 1
+        node.height = max([self.height(node.left), self.height(node.right)]) + 1
+
+    def balance_factor(self, node: AVLTreeNode) -> int:
+        """
+        获取结点平衡因子
+        Args:
+            node: 要获取平衡因子的结点
+
+        Returns: 平衡因子
+
+        """
+        # 空结点平衡因子为 0
+        if node is None:
+            return 0
+        # 结点平衡因子 = 左子树高度 - 右子树高度
+        return self.height(node.left) - self.height(node.right)
+
+    def __right_rotate(self, node: AVLTreeNode) -> AVLTreeNode:
+        child = node.left
+        grand_child = child.right
+        # 以 child 为原点，将 node 向右旋转
+        child.right = node
+        node.left = grand_child
+        # 更新结点高度
+        self.__update_height(node)
+        self.__update_height(child)
+        # 返回旋转后子树的根节点
+        return child
+
+    def __left_rotate(self, node: AVLTreeNode) -> AVLTreeNode:
+        child = node.right
+        grand_child = child.left
+        # 以 child 为原点，将 node 向左旋转
+        child.left = node
+        node.right = grand_child
+        # 更新结点高度
+        self.__update_height(node)
+        self.__update_height(child)
+        # 返回旋转后子树的根节点
+        return child
+
+    def rotate(self, node: AVLTreeNode):
+        """
+        执行旋转操作，使该子树重新恢复平衡
+        Args:
+            node: 要旋转的根结点
+
+        Returns: 旋转后的根结点
+
+        """
+        # 获取结点 node 的平衡因子
+        balance_factor = self.balance_factor(node)
+        # 左偏树
+        if balance_factor > 1:
+            if self.balance_factor(node.left) >= 0:
+                # 右旋
+                return self.__right_rotate(node)
+            else:
+                # 先左旋后右旋
+                node.left = self.__left_rotate(node.left)
+                return self.__right_rotate(node)
+        # 右偏树
+        elif balance_factor < -1:
+            if self.balance_factor(node.right) <= 0:
+                # 左旋
+                return self.__left_rotate(node)
+            else:
+                # 先右旋后左旋
+                node.right = self.__right_rotate(node.right)
+                return self.__left_rotate(node)
+        # 平衡树，无需旋转，直接返回
+        return node
+
+    def insert(self, val) -> AVLTreeNode:
+        """
+        插入结点
+        Args:
+            val: 结点的值
+
+        Returns:
+            node: 插入结点后的根结点
+        """
+        self.root = self.insert_helper(self.root, val)
+        return self.root
+
+    def insert_helper(
+        self, node: typing.Optional[AVLTreeNode], val: int
+    ) -> AVLTreeNode:
+        """
+        递归插入结点（辅助函数）
+        Args:
+            node: 要插入的根结点
+            val: 要插入的结点的值
+
+        Returns: 插入结点后的根结点
+
+        """
+        if node is None:
+            return AVLTreeNode(val)
+        # 1. 查找插入位置，并插入结点
+        if val < node.val:
+            node.left = self.insert_helper(node.left, val)
+        elif val > node.val:
+            node.right = self.insert_helper(node.right, val)
+        else:
+            # 重复结点不插入，直接返回
+            return node
+        # 更新结点高度
+        self.__update_height(node)
+        # 2. 执行旋转操作，使该子树重新恢复平衡
+        return self.rotate(node)
+
+    def remove(self, val: int):
+        """
+        删除结点
+        Args:
+            val: 要删除的结点的值
+
+        Returns:
+
+        """
+        root = self.remove_helper(self.root, val)
+        return root
+
+    def remove_helper(
+        self, node: typing.Optional[AVLTreeNode], val: int
+    ) -> typing.Optional[AVLTreeNode]:
+        """
+        递归删除结点（辅助函数）
+        Args:
+            node:  删除的起始结点
+            val: 要删除的结点的值
+
+        Returns: 删除目标结点后的起始结点
+
+        """
+        if node is None:
+            return None
+        # 1. 查找结点，并删除之
+        if val < node.val:
+            node.left = self.remove_helper(node.left, val)
+        elif val > node.val:
+            node.right = self.remove_helper(node.right, val)
+        else:
+            if node.left is None or node.right is None:
+                child = node.left or node.right
+                # 子结点数量 = 0 ，直接删除 node 并返回
+                if child is None:
+                    return None
+                # 子结点数量 = 1 ，直接删除 node
+                else:
+                    node = child
+            else:  # 子结点数量 = 2 ，则将中序遍历的下个结点删除，并用该结点替换当前结点
+                temp = self.min_node(node.right)
+                node.right = self.remove_helper(node.right, temp.val)
+                node.val = temp.val
+        # 更新结点高度
+        self.__update_height(node)
+        # 2. 执行旋转操作，使该子树重新恢复平衡
+        return self.rotate(node)
+
+    def min_node(
+        self, node: typing.Optional[AVLTreeNode]
+    ) -> typing.Optional[AVLTreeNode]:
+        # 获取最小结点
+        if node is None:
+            return None
+        # 循环访问左子结点，直到叶结点时为最小结点，跳出
+        while node.left is not None:
+            node = node.left
+        return node
+
+    def search(self, val: int):
+        cur = self.root
+        while cur is not None:
+            if cur.val < val:
+                cur = cur.right
+            elif cur.val > val:
+                cur = cur.left
+            else:
+                break
+        return cur
+
+
+if __name__ == "__main__":
+
+    def test_insert(tree: AVLTree, val: int):
+        tree.insert(val)
+        print("\n插入结点 {} 后，AVL 树为".format(val))
+        print_tree(tree.root)
+
+    def test_remove(tree: AVLTree, val: int):
+        tree.remove(val)
+        print("\n删除结点 {} 后，AVL 树为".format(val))
+        print_tree(tree.root)
+
+    # 初始化空 AVL 树
+    avl_tree = AVLTree()
+
+    # 插入结点
+    # 请关注插入结点后，AVL 树是如何保持平衡的
+    test_insert(avl_tree, 1)
+    test_insert(avl_tree, 2)
+    test_insert(avl_tree, 3)
+    test_insert(avl_tree, 4)
+    test_insert(avl_tree, 5)
+    test_insert(avl_tree, 8)
+    test_insert(avl_tree, 7)
+    test_insert(avl_tree, 9)
+    test_insert(avl_tree, 10)
+    test_insert(avl_tree, 6)
+
+    # 插入重复结点
+    test_insert(avl_tree, 7)
+
+    # 删除结点
+    # 请关注删除结点后，AVL 树是如何保持平衡的
+    test_remove(avl_tree, 8)  # 删除度为 0 的结点
+    test_remove(avl_tree, 5)  # 删除度为 1 的结点
+    test_remove(avl_tree, 4)  # 删除度为 2 的结点
+
+    result_node = avl_tree.search(7)
+    print("\n查找到的结点对象为 {}，结点值 = {}".format(result_node, result_node.val))

codes/python/chapter_tree/binary_search_tree.py

@@ -1,10 +1,173 @@
-'''
+"""
 File: binary_search_tree.py
 Created Time: 2022-11-25
 Author: Krahets (krahets@163.com)
-'''
+"""
 
 import sys, os.path as osp
+
 sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
 from include import *
 
+
+class BinarySearchTree:
+    """
+    二叉搜索树
+    """
+
+    def __init__(self, nums) -> None:
+        nums.sort()
+        self.__root = self.buildTree(nums, 0, len(nums) - 1)
+
+    def buildTree(self, nums, start_index, end_index):
+        if start_index > end_index:
+            return None
+        mid = (start_index + end_index) // 2
+        root = TreeNode(nums[mid])
+        root.left = self.buildTree(
+            nums=nums, start_index=start_index, end_index=mid - 1
+        )
+        root.right = self.buildTree(nums=nums, start_index=mid + 1, end_index=end_index)
+        return root
+
+    def get_root(self):
+        return self.__root
+
+    def search(self, num):
+        """
+        查找结点
+        """
+        cur = self.get_root()
+        # 循环查找，越过叶结点后跳出
+        while cur is not None:
+            # 目标结点在 root 的右子树中
+            if cur.val < num:
+                cur = cur.right
+            # 目标结点在 root 的左子树中
+            elif cur.val > num:
+                cur = cur.left
+            # 找到目标结点，跳出循环
+            else:
+                break
+        return cur
+
+    def insert(self, num):
+        """
+        插入结点
+        """
+        root = self.get_root()
+        # 若树为空，直接提前返回
+        if root is None:
+            return None
+
+        cur = root
+        pre = None
+
+        # 循环查找，越过叶结点后跳出
+        while cur is not None:
+            # 找到重复结点，直接返回
+            if cur.val == num:
+                return None
+            pre = cur
+
+            if cur.val < num:  # 插入位置在 root 的右子树中
+                cur = cur.right
+            else:  # 插入位置在 root 的左子树中
+                cur = cur.left
+
+        # 插入结点 val
+        node = TreeNode(num)
+        if pre.val < num:
+            pre.right = node
+        else:
+            pre.left = node
+        return node
+
+    def remove(self, num):
+        """
+        删除结点
+        """
+        root = self.get_root()
+        # 若树为空，直接提前返回
+        if root is None:
+            return None
+
+        cur = root
+        pre = None
+
+        # 循环查找，越过叶结点后跳出
+        while cur is not None:
+            # 找到待删除结点，跳出循环
+            if cur.val == num:
+                break
+            pre = cur
+            if cur.val < num:  # 待删除结点在 root 的右子树中
+                cur = cur.right
+            else:  # 待删除结点在 root 的左子树中
+                cur = cur.left
+
+        # 若无待删除结点，则直接返回
+        if cur is None:
+            return None
+
+        # 子结点数量 = 0 or 1
+        if cur.left is None or cur.right is None:
+            # 当子结点数量 = 0 / 1 时， child = null / 该子结点
+            child = cur.left or cur.right
+            # 删除结点 cur
+            if pre.left == cur:
+                pre.left = child
+            else:
+                pre.right = child
+        # 子结点数量 = 2
+        else:
+            # 获取中序遍历中 cur 的下一个结点
+            nex = self.min(cur.right)
+            tmp = nex.val
+            # 递归删除结点 nex
+            self.remove(nex.val)
+            # 将 nex 的值复制给 cur
+            cur.val = tmp
+        return cur
+
+    def min(self, root):
+        """
+        获取最小结点
+        """
+        if root is None:
+            return root
+
+        # 循环访问左子结点，直到叶结点时为最小结点，跳出
+        while root.left is not None:
+            root = root.left
+        return root
+
+
+if __name__ == "__main__":
+    # 初始化二叉搜索树
+    nums = list(range(1, 16))
+    bst = BinarySearchTree(nums=nums)
+    print("\n初始化的二叉树为\n")
+    print_tree(bst.get_root())
+
+    # 查找结点
+    node = bst.search(5)
+    print("\n查找到的结点对象为: {}，结点值 = {}".format(node, node.val))
+
+    # 插入结点
+    ndoe = bst.insert(16)
+    print("\n插入结点 16 后，二叉树为\n")
+    print_tree(bst.get_root())
+
+    # 删除结点
+    bst.remove(1)
+    print("\n删除结点 1 后，二叉树为\n")
+    print_tree(bst.get_root())
+
+    bst.remove(2)
+    print("\n删除结点 2 后，二叉树为\n")
+    print_tree(bst.get_root())
+
+    bst.remove(4)
+    print("\n删除结点 4 后，二叉树为\n")
+    print_tree(bst.get_root())

codes/python/chapter_tree/binary_tree.py

@@ -1,10 +1,38 @@
-'''
+"""
 File: binary_tree.py
 Created Time: 2022-11-25
 Author: Krahets (krahets@163.com)
-'''
+"""
 
 import sys, os.path as osp
+
 sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
 from include import *
 
+
+""" Driver Code """
+if __name__ == "__main__":
+    # 初始化二叉树
+    # 初始化节点
+    n1 = TreeNode(val=1)
+    n2 = TreeNode(val=2)
+    n3 = TreeNode(val=3)
+    n4 = TreeNode(val=4)
+    n5 = TreeNode(val=5)
+    n1.left = n2
+    n1.right = n3
+    n2.left = n4
+    n2.right = n5
+    print_tree(n1)
+
+    # 插入与删除结点
+    P = TreeNode(0)
+
+    # 在 n1 -> n2 中间插入节点 P
+    n1.left = P
+    P.left = n2
+    print_tree(n1)
+
+    # 删除结点
+    n1.left = n2
+    print_tree(n1)

codes/python/chapter_tree/binary_tree_bfs.py

@@ -1,10 +1,45 @@
-'''
+"""
 File: binary_tree_bfs.py
 Created Time: 2022-11-25
 Author: Krahets (krahets@163.com)
-'''
+"""
 
 import sys, os.path as osp
 sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
 from include import *
 
+
+def hierOrder(root):
+    # 初始化队列，加入根结点
+    queue = collections.deque()
+    queue.append(root)
+    # 初始化一个列表，用于保存遍历序列
+    result = []
+    while queue:
+        # 队列出队
+        node = queue.popleft()
+        # 保存节点值
+        result.append(node.val)
+        if node.left is not None:
+            # 左子结点入队
+            queue.append(node.left)
+        if node.right is not None:
+            # 右子结点入队
+            queue.append(node.right)
+    return result
+
+
+""" Driver Code """
+if __name__ == "__main__":
+    # 初始化二叉树
+    # 这里借助了一个从数组直接生成二叉树的函数
+    root = list_to_tree(
+        arr=[1, 2, 3, 4, 5, 6, 7, None, None, None, None, None, None, None, None]
+    )
+    print("\n初始化二叉树\n")
+    print_tree(root)
+
+    # 层序遍历
+    result = hierOrder(root)
+    print("\n层序遍历的结点打印序列 = ", result)
+    assert result == [1, 2, 3, 4, 5, 6, 7]

codes/python/chapter_tree/binary_tree_dfs.py

@@ -1,10 +1,80 @@
-'''
+"""
 File: binary_tree_dfs.py
 Created Time: 2022-11-25
 Author: Krahets (krahets@163.com)
-'''
+"""
 
 import sys, os.path as osp
 sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
 from include import *
 
+
+result = []
+
+
+def preOrder(root):
+    """
+    前序遍历二叉树
+    """
+    if root is None:
+        return
+
+    # 访问优先级：根结点 -> 左子树 -> 右子树
+    result.append(root.val)
+    preOrder(root=root.left)
+    preOrder(root=root.right)
+
+
+def inOrder(root):
+    """
+    中序遍历二叉树
+    """
+    if root is None:
+        return
+
+    # 访问优先级：左子树 -> 根结点 -> 右子树
+    inOrder(root=root.left)
+    result.append(root.val)
+    inOrder(root=root.right)
+
+
+def postOrder(root):
+    """
+    后序遍历二叉树
+    """
+    if root is None:
+        return
+
+    # 访问优先级：左子树 -> 右子树 -> 根结点
+    postOrder(root=root.left)
+    postOrder(root=root.right)
+    result.append(root.val)
+
+
+""" Driver Code """
+if __name__ == "__main__":
+    # 初始化二叉树
+    # 这里借助了一个从数组直接生成二叉树的函数
+    root = list_to_tree(
+        arr=[1, 2, 3, 4, 5, 6, 7, None, None, None, None, None, None, None, None]
+    )
+    print("\n初始化二叉树\n")
+    print_tree(root)
+
+    # 前序遍历
+    result = []
+    preOrder(root)
+    print("\n前序遍历的结点打印序列 = ", result)
+    assert result == [1, 2, 4, 5, 3, 6, 7]
+
+    # 中序遍历
+    result = []
+    inOrder(root)
+    print("\n中序遍历的结点打印序列 = ", result)
+    assert result == [4, 2, 5, 1, 6, 3, 7]
+
+    # 后序遍历
+    result = []
+    postOrder(root)
+    print("\n后序遍历的结点打印序列 = ", result)
+    assert result == [4, 5, 2, 6, 7, 3, 1]

codes/python/include/binary_tree.py

@@ -9,7 +9,7 @@ import collections
 class TreeNode:
     """Definition for a binary tree node
     """    
-    def __init__(self, val=0, left=None, right=None):
+    def __init__(self, val=None, left=None, right=None):
         self.val = val
         self.left = left
         self.right = right

docs/chapter_tree/avl_tree.md

@@ -48,7 +48,24 @@ G. M. Adelson-Velsky 和 E. M. Landis 在其 1962 年发表的论文 "An algorit
 === "Python"
 
     ```python title="avl_tree.py"
-    
+    class AVLTreeNode:
+        def __init__(
+                self,
+                val=None,
+                height: int = 0,
+                left: typing.Optional["AVLTreeNode"] = None,
+                right: typing.Optional["AVLTreeNode"] = None
+        ):
+            self.val = val
+            self.height = height
+            self.left = left
+            self.right = right
+
+        def __str__(self):
+            val = self.val
+            left_val = self.left.val if self.left else None
+            right_val = self.right.val if self.right else None
+            return "<AVLTreeNode: {}, leftAVLTreeNode: {}, rightAVLTreeNode: {}>".format(val, left_val, right_val)
     ```
 
 === "Go"
@@ -108,7 +125,31 @@ G. M. Adelson-Velsky 和 E. M. Landis 在其 1962 年发表的论文 "An algorit
 === "Python"
 
     ```python title="avl_tree.py"
-    
+    def height(node: typing.Optional[AVLTreeNode]) -> int:
+        """
+        获取结点高度
+        Args:
+            node:起始结点 
+
+        Returns: 高度 or -1
+
+        """
+        # 空结点高度为 -1 ，叶结点高度为 0
+        if node is not None:
+            return node.height
+        return -1
+
+    def update_height(node: AVLTreeNode):
+        """
+        更新结点高度
+        Args:
+            node: 要更新高度的结点
+
+        Returns: None
+
+        """
+        # 结点高度等于最高子树高度 + 1
+        node.height = max([height(node.left), height(node.right)]) + 1
     ```
 
 === "Go"
@@ -166,7 +207,20 @@ G. M. Adelson-Velsky 和 E. M. Landis 在其 1962 年发表的论文 "An algorit
 === "Python"
 
     ```python title="avl_tree.py"
-    
+    def balance_factor(node: AVLTreeNode) -> int:
+        """
+        获取结点平衡因子
+        Args:
+            node: 要获取平衡因子的结点
+
+        Returns: 平衡因子
+
+        """
+        # 空结点平衡因子为 0
+        if node is None:
+            return 0
+        # 结点平衡因子 = 左子树高度 - 右子树高度
+        return height(node.left) - height(node.right)
     ```
 
 === "Go"
@@ -255,7 +309,17 @@ AVL 树的独特之处在于「旋转 Rotation」的操作，其可 **在不影
 === "Python"
 
     ```python title="avl_tree.py"
-    
+    def rightRotate(node: AVLTreeNode):
+        child = node.left
+        grand_child = child.right
+        # 以 child 为原点，将 node 向右旋转
+        child.right = node
+        node.left = grand_child
+        # 更新结点高度
+        update_height(node)
+        update_height(child)
+        # 返回旋转后子树的根节点
+        return child
     ```
 
 === "Go"
@@ -323,7 +387,17 @@ AVL 树的独特之处在于「旋转 Rotation」的操作，其可 **在不影
 === "Python"
 
     ```python title="avl_tree.py"
-    
+    def leftRotate(node: AVLTreeNode):
+        child = node.right
+        grand_child = child.left
+        # 以 child 为原点，将 node 向左旋转
+        child.left = node
+        node.right = grand_child
+        # 更新结点高度
+        update_height(node)
+        update_height(child)
+        # 返回旋转后子树的根节点
+        return child
     ```
 
 === "Go"
@@ -432,7 +506,37 @@ AVL 树的独特之处在于「旋转 Rotation」的操作，其可 **在不影
 === "Python"
 
     ```python title="avl_tree.py"
-    
+    def rotate(node: AVLTreeNode):
+        """
+        执行旋转操作，使该子树重新恢复平衡
+        Args:
+            node: 要旋转的根结点
+
+        Returns: 旋转后的根结点
+
+        """
+        # 获取结点 node 的平衡因子
+        factor = balance_factor(node)
+        # 左偏树
+        if factor > 1:
+            if balance_factor(node.left) >= 0:
+                # 右旋
+                return right_rotate(node)
+            else:
+                # 先左旋后右旋
+                node.left = left_rotate(node.left)
+                return right_rotate(node)
+        # 右偏树
+        elif factor < -1:
+            if balance_factor(node.right) <= 0:
+                # 左旋
+                return left_rotate(node)
+            else:
+                # 先右旋后左旋
+                node.right = right_rotate(node.right)
+                return left_rotate(node)
+        # 平衡树，无需旋转，直接返回
+        return node
     ```
 
 === "Go"
@@ -507,7 +611,42 @@ AVL 树的独特之处在于「旋转 Rotation」的操作，其可 **在不影
 === "Python"
 
     ```python title="avl_tree.py"
-    
+    def insert(val) -> AVLTreeNode:
+        """
+        插入结点
+        Args:
+            val: 结点的值 
+
+        Returns:
+            node: 插入结点后的根结点
+        """
+        root = insert_helper(root, val)
+        return root
+
+    def insert_helper(node: typing.Optional[AVLTreeNode], val: int) -> AVLTreeNode:
+        """
+        递归插入结点（辅助函数）
+        Args:
+            node: 要插入的根结点
+            val: 要插入的结点的值
+
+        Returns: 插入结点后的根结点
+
+        """
+        if node is None:
+            return AVLTreeNode(val)
+        # 1. 查找插入位置，并插入结点
+        if val < node.val:
+            node.left = insert_helper(node.left, val)
+        elif val > node.val:
+            node.right = insert_helper(node.right, val)
+        else:
+            # 重复结点不插入，直接返回
+            return node
+        # 更新结点高度
+        update_height(node)
+        # 2. 执行旋转操作，使该子树重新恢复平衡
+        return rotate(node)
     ```
 
 === "Go"
@@ -604,7 +743,62 @@ AVL 树的独特之处在于「旋转 Rotation」的操作，其可 **在不影
 === "Python"
 
     ```python title="avl_tree.py"
-    
+    def remove(val: int):
+        """
+        删除结点
+        Args:
+            val: 要删除的结点的值
+
+        Returns:
+
+        """
+        root = remove_helper(root, val)
+        return root
+
+    def remove_helper(node: typing.Optional[AVLTreeNode], val: int) -> typing.Optional[AVLTreeNode]:
+        """
+        递归删除结点（辅助函数）
+        Args:
+            node:  删除的起始结点
+            val: 要删除的结点的值
+
+        Returns: 删除目标结点后的起始结点
+
+        """
+        if node is None:
+            return None
+        # 1. 查找结点，并删除之
+        if val < node.val:
+            node.left = remove_helper(node.left, val)
+        elif val > node.val:
+            node.right = remove_helper(node.right, val)
+        else:
+            if node.left is None or node.right is None:
+                child = node.left or node.right
+                # 子结点数量 = 0 ，直接删除 node 并返回
+                if child is None:
+                    return None
+                # 子结点数量 = 1 ，直接删除 node
+                else:
+                    node = child
+            else: # 子结点数量 = 2 ，则将中序遍历的下个结点删除，并用该结点替换当前结点
+                temp = min_node(node.right)
+                node.right = remove_helper(node.right, temp.val)
+                node.val = temp.val
+        # 更新结点高度
+        update_height(node)
+        # 2. 执行旋转操作，使该子树重新恢复平衡
+        return rotate(node)
+
+
+    def min_node(node: typing.Optional[AVLTreeNode]) -> typing.Optional[AVLTreeNode]:
+        # 获取最小结点
+        if node is None:
+            return None
+        # 循环访问左子结点，直到叶结点时为最小结点，跳出
+        while node.left is not None:
+            node = node.left
+        return node
     ```
 
 === "Go"

docs/chapter_tree/binary_search_tree.md

@@ -82,6 +82,23 @@ comments: true
 === "Python"
 
     ```python title="binary_search_tree.py"
+    def search(self, num):
+        """
+        查找结点
+        """
+        cur = self.get_root()
+        # 循环查找，越过叶结点后跳出
+        while cur is not None:
+            # 目标结点在 root 的右子树中
+            if cur.val < num:
+                cur = cur.right
+            # 目标结点在 root 的左子树中
+            elif cur.val > num:
+                cur = cur.left
+            # 找到目标结点，跳出循环
+            else:
+                break
+        return cur
 
     ```
 
@@ -228,7 +245,37 @@ comments: true
 === "Python"
 
     ```python title="binary_search_tree.py"
-
+    def insert(self, num):
+        """
+        插入结点
+        """
+        root = self.get_root()
+        # 若树为空，直接提前返回
+        if root is None:
+            return None
+
+        cur = root
+        pre = None
+
+        # 循环查找，越过叶结点后跳出
+        while cur is not None:
+            # 找到重复结点，直接返回
+            if cur.val == num:
+                return None
+            pre = cur
+
+            if cur.val < num:  # 插入位置在 root 的右子树中
+                cur = cur.right
+            else:  # 插入位置在 root 的左子树中
+                cur = cur.left
+
+        # 插入结点 val
+        node = TreeNode(num)
+        if pre.val < num:
+            pre.right = node
+        else:
+            pre.left = node
+        return node
     ```
 
 === "Go"
@@ -483,7 +530,64 @@ comments: true
 === "Python"
 
     ```python title="binary_search_tree.py"
+    def remove(self, num):
+        """
+        删除结点
+        """
+        root = self.get_root()
+        # 若树为空，直接提前返回
+        if root is None:
+            return None
+        
+        cur = root
+        pre = None
+
+        # 循环查找，越过叶结点后跳出
+        while cur is not None:
+            # 找到待删除结点，跳出循环
+            if cur.val == num:
+                break
+            pre = cur
+            if cur.val < num:  # 待删除结点在 root 的右子树中
+                cur = cur.right
+            else:  # 待删除结点在 root 的左子树中
+                cur = cur.left
+
+        # 若无待删除结点，则直接返回
+        if cur is None:
+            return None
+
+        # 子结点数量 = 0 or 1
+        if cur.left is None or cur.right is None:
+            # 当子结点数量 = 0 / 1 时， child = null / 该子结点
+            child = cur.left or cur.right
+            # 删除结点 cur
+            if pre.left == cur:
+                pre.left = child
+            else:
+                pre.right = child
+        # 子结点数量 = 2
+        else:
+            # 获取中序遍历中 cur 的下一个结点
+            nex = self.min(cur.right)
+            tmp = nex.val
+            # 递归删除结点 nex
+            self.remove(nex.val)
+            # 将 nex 的值复制给 cur
+            cur.val =  tmp
+        return cur
 
+    def min(self, root):
+        """
+        获取最小结点
+        """
+        if root is None:
+            return root
+
+        # 循环访问左子结点，直到叶结点时为最小结点，跳出
+        while root.left is not None:
+            root = root.left
+        return root
     ```
 
 === "Go"

docs/chapter_tree/binary_tree.md

@@ -33,9 +33,9 @@ comments: true
 === "Python"
 
     ```python title=""
-    """ 链表结点类 """
     class TreeNode:
-        def __init__(self, val=0, left=None, right=None):
+        """ 链表结点类 """
+        def __init__(self, val=None, left=None, right=None):
             self.val = val      # 结点值
             self.left = left    # 左子结点指针
             self.right = right  # 右子结点指针
@@ -190,7 +190,18 @@ comments: true
 === "Python"
 
     ```python title="binary_tree.py"
-
+    # 初始化二叉树
+    # 初始化节点
+    n1 = TreeNode(val=1)
+    n2 = TreeNode(val=2)
+    n3 = TreeNode(val=3)
+    n4 = TreeNode(val=4)
+    n5 = TreeNode(val=5)
+    # 构建引用指向（即指针）
+    n1.left = n2
+    n1.right = n3
+    n2.left = n4
+    n2.right = n5
     ```
 
 === "Go"
@@ -288,7 +299,13 @@ comments: true
 === "Python"
 
     ```python title="binary_tree.py"
-
+    # 插入与删除结点
+    p = TreeNode(0)
+    # 在 n1 -> n2 中间插入结点 P
+    n1.left = p
+    p.left = n2
+    # 删除节点 P
+    n1.left = n2
     ```
 
 === "Go"
@@ -406,7 +423,24 @@ comments: true
 === "Python"
 
     ```python title="binary_tree_bfs.py"
-
+    def hierOrder(root):
+        # 初始化队列，加入根结点
+        queue = collections.deque()
+        queue.append(root)
+        # 初始化一个列表，用于保存遍历序列
+        result = []
+        while queue:
+            # 队列出队
+            node = queue.popleft()
+            # 保存节点值
+            result.append(node.val)
+            if node.left is not None:
+                # 左子结点入队
+                queue.append(node.left)
+            if node.right is not None:
+                # 右子结点入队
+                queue.append(node.right)
+        return result
     ```
 
 === "Go"
@@ -578,7 +612,43 @@ comments: true
 === "Python"
 
     ```python title="binary_tree_dfs.py"
+    def preOrder(root):
+        """
+        前序遍历二叉树
+        """
+        if root is None:
+            return
+        
+        # 访问优先级：根结点 -> 左子树 -> 右子树
+        result.append(root.val)
+        preOrder(root=root.left)
+        preOrder(root=root.right)
+
+
+    def inOrder(root):
+        """
+        中序遍历二叉树
+        """
+        if root is None:
+            return
+
+        # 访问优先级：左子树 -> 根结点 -> 右子树
+        inOrder(root=root.left)
+        result.append(root.val)
+        inOrder(root=root.right)
 
+
+    def postOrder(root):
+        """
+        后序遍历二叉树
+        """
+        if root is None:
+            return
+        
+        # 访问优先级：左子树 -> 右子树 -> 根结点
+        postOrder(root=root.left)
+        postOrder(root=root.right)
+        result.append(root.val)
     ```
 
 === "Go"