二叉树算法精要：从基础到实战

二叉树基础结构

二叉树的每个节点包含三个部分：数据域、左子节点指针和右子节点指针。结构定义通常使用如下代码表示：

struct TreeNode {
    int val;
    struct TreeNode *left;
    struct TreeNode *right;
};

在面向对象语言中可采用类实现：

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

核心遍历方法

递归式深度优先遍历

# 前序遍历
def preorder(root):
    if not root: return
    print(root.val)
    preorder(root.left)
    preorder(root.right)

# 中序遍历
def inorder(root):
    if not root: return
    inorder(root.left)
    print(root.val)
    inorder(root.right)

# 后序遍历
def postorder(root):
    if not root: return
    postorder(root.left)
    postorder(root.right)
    print(root.val)

迭代式遍历（使用栈）

def preorder_iter(root):
    stack = []
    while root or stack:
        while root:
            print(root.val)
            stack.append(root)
            root = root.left
        root = stack.pop()
        root = root.right

层序遍历（BFS）

from collections import deque
def level_order(root):
    q = deque([root]) if root else deque()
    while q:
        node = q.popleft()
        print(node.val)
        if node.left: q.append(node.left)
        if node.right: q.append(node.right)

关键操作接口实现

计算树深度

def max_depth(root):
    if not root: return 0
    return 1 + max(max_depth(root.left), max_depth(root.right))

镜像对称判断

def is_symmetric(root):
    def mirror(left, right):
        if not left and not right: return True
        if not left or not right: return False
        return left.val == right.val and mirror(left.left, right.right) and mirror(left.right, right.left)
    return mirror(root.left, root.right) if root else True

OJ 实战技巧

路径总和问题

def has_path_sum(root, target):
    if not root: return False
    if not root.left and not root.right:
        return target == root.val
    return has_path_sum(root.left, target - root.val) or has_path_sum(root.right, target - root.val)

最近公共祖先

def lowest_common_ancestor(root, p, q):
    if not root or root == p or root == q: return root
    left = lowest_common_ancestor(root.left, p, q)
    right = lowest_common_ancestor(root.right, p, q)
    if left and right: return root
    return left if left else right

二叉搜索树验证

def is_valid_bst(root):
    def validate(node, min_val, max_val):
        if not node: return True
        if node.val <= min_val or node.val >= max_val:
            return False
        return validate(node.left, min_val, node.val) and validate(node.right, node.val, max_val)
    return validate(root, float('-inf'), float('inf'))

性能优化要点

递归转迭代可避免栈溢出问题，层序遍历使用双端队列比列表更高效。对于平衡二叉树问题，提前终止条件的设置能显著减少计算量。在路径类问题中，回溯法的剪枝操作可降低时间复杂度。

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