Merge branch 'master' of github.com:krahets/hello-algo

docs/chapter_tree/avl_tree.md

@@ -48,7 +48,21 @@ 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"
-    
+    """ AVL 树结点类 """
+    class TreeNode:
+        def __init__(self, val=None, left=None, right=None):
+            self.val = val  # 结点值
+            self.height = 0  # 结点高度, avl 树会用到
+            self.left = left  # 左子结点引用
+            self.right = right  # 右子结点引用
+
+        def __str__(self):  # 直接print时会好看一点
+            val = self.val
+            left_node_val = self.left.val if self.left else None
+            right_node_val = self.right.val if self.right else None
+            return "<TreeNode: {}, leftTreeNode: {}, rightTreeNode: {}>".format(val, left_node_val, right_node_val)
+
+        __repr__ = __str__
     ```
 
 === "Go"
@@ -115,7 +129,17 @@ 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"
-    
+    """ 获取结点高度 """
+    def height(self, node: typing.Optional[TreeNode]) -> int:
+        # 空结点高度为 -1 ，叶结点高度为 0
+        if node is not None:
+            return node.height
+        return -1
+
+    """ 更新结点高度 """
+    def __update_height(self, node: TreeNode):
+        # 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
     ```
 
 === "Go"
@@ -185,7 +209,13 @@ 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"
-    
+    """ 获取平衡因子 """
+    def balance_factor(self, node: TreeNode) -> int:
+        # 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)
     ```
 
 === "Go"
@@ -281,7 +311,18 @@ 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"
-    
+    """ 右旋操作 """
+    def __right_rotate(self, node: TreeNode) -> TreeNode:
+        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
     ```
 
 === "Go"
@@ -363,7 +404,18 @@ 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"
-    
+    """ 左旋操作 """
+    def __left_rotate(self, node: TreeNode) -> TreeNode:
+        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
     ```
 
 === "Go"
@@ -485,7 +537,30 @@ 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"
-    
+    """ 执行旋转操作，使该子树重新恢复平衡 """
+    def __rotate(self, node: TreeNode) -> TreeNode:
+        # 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
     ```
 
 === "Go"
@@ -597,7 +672,27 @@ 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"
-    
+    """ 插入结点 """
+    def insert(self, val) -> TreeNode:
+        self.root = self.__insert_helper(self.root, val)
+        return self.root
+
+    """ 递归插入结点（辅助函数）"""
+    def __insert_helper(self, node: typing.Optional[TreeNode], val: int) -> TreeNode:
+        if node is None:
+            return TreeNode(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)
     ```
 
 === "Go"
@@ -717,7 +812,46 @@ 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"
-    
+    """ 删除结点 """
+    def remove(self, val: int):
+        root = self.__remove_helper(self.root, val)
+        return root
+
+    """ 递归删除结点（辅助函数） """
+    def __remove_helper(self, node: typing.Optional[TreeNode], val: int) -> typing.Optional[TreeNode]:
+        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)
+
+    """ 获取最小结点 """
+    def min_node(self, node: typing.Optional[TreeNode]) -> typing.Optional[TreeNode]:
+        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

@@ -82,7 +82,21 @@ comments: true
 === "Python"
 
     ```python title="binary_search_tree.py"
-
+    """ 查找结点 """
+    def search(self, num: int) -> typing.Optional[TreeNode]:
+        cur = self.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
     ```
 
 === "Go"
@@ -244,7 +258,35 @@ comments: true
 === "Python"
 
     ```python title="binary_search_tree.py"
+    """ 插入结点 """
+    def insert(self, num: int) -> typing.Optional[TreeNode]:
+        root = self.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"
@@ -525,7 +567,60 @@ comments: true
 === "Python"
 
     ```python title="binary_search_tree.py"
+    """ 删除结点 """
+    def remove(self, num: int) -> typing.Optional[TreeNode]:
+        root = self.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
+
+    """ 获取最小结点 """
+    def min(self, root: typing.Optional[TreeNode]) -> typing.Optional[TreeNode]:
+        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

@@ -35,7 +35,7 @@ comments: true
     ```python title=""
     """ 链表结点类 """
     class TreeNode:
-        def __init__(self, val=0, left=None, right=None):
+        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
@@ -175,7 +175,18 @@ 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"
@@ -284,7 +295,13 @@ 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"
@@ -437,7 +454,7 @@ comments: true
 === "Python"
 
     ```python title=""
-    “”“ 二叉树的数组表示 ”“”
+    """ 二叉树的数组表示 """
     # 直接使用 None 来表示空位
     tree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]
     ```

docs/chapter_tree/binary_tree_traversal.md

@@ -51,12 +51,12 @@ comments: true
         vector<int> vec;
         while (!queue.empty()) {
             TreeNode* node = queue.front();
-            queue.pop();  // 队列出队
-            vec.push_back(node->val);            // 保存结点
+            queue.pop();                 // 队列出队
+            vec.push_back(node->val);    // 保存结点
             if (node->left != nullptr)
-                queue.push(node->left);    // 左子结点入队
+                queue.push(node->left);  // 左子结点入队
             if (node->right != nullptr)
-                queue.push(node->right);   // 右子结点入队
+                queue.push(node->right); // 右子结点入队
         }
         return vec;
     }
@@ -65,7 +65,21 @@ comments: true
 === "Python"
 
     ```python title="binary_tree_bfs.py"
-    
+    """ 层序遍历 """
+    def hier_order(root: TreeNode):
+        # 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
+        res = []
+        while queue:
+            node = queue.popleft()       # 队列出队
+            res.append(node.val)         # 保存节点值
+            if node.left is not None:
+                queue.append(node.left)  # 左子结点入队
+            if node.right is not None:
+                queue.append(node.right) # 右子结点入队
+        return res
     ```
 
 === "Go"
@@ -256,7 +270,32 @@ comments: true
 === "Python"
 
     ```python title="binary_tree_dfs.py"
-    
+    """ 前序遍历 """
+    def pre_order(root: typing.Optional[TreeNode]):
+        if root is None:
+            return
+        # 访问优先级：根结点 -> 左子树 -> 右子树
+        res.append(root.val)
+        pre_order(root=root.left)
+        pre_order(root=root.right)
+
+    """ 中序遍历 """
+    def in_order(root: typing.Optional[TreeNode]):
+        if root is None:
+            return
+        # 访问优先级：左子树 -> 根结点 -> 右子树
+        in_order(root=root.left)
+        res.append(root.val)
+        in_order(root=root.right)
+
+    """ 后序遍历 """
+    def post_order(root: typing.Optional[TreeNode]):
+        if root is None:
+            return
+        # 访问优先级：左子树 -> 右子树 -> 根结点
+        post_order(root=root.left)
+        post_order(root=root.right)
+        res.append(root.val)
     ```
 
 === "Go"
@@ -402,7 +441,6 @@ comments: true
         postOrder(root.right);
         list.Add(root.val);
     }
-    
     ```
 
 !!! note