PAT A1043：Is It a Binary Search Tree（二叉搜索树）

如何构建树？如何遍历树？镜像树的遍历指的是什么？

本文作者：MowChan · 发布日期：2021/01/17 · 阅读时长：8 分钟

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题目内容

A Binary Search Tree (BST) is recursively defined as a binary tree which has the following properties:

  • The left subtree of a node contains only nodes with keys less than the node’s key.
  • The right subtree of a node contains only nodes with keys greater than or equal to the node’s key.
  • Both the left and right subtrees must also be binary search trees.

If we swap the left and right subtrees of every node, then the resulting tree is called the Mirror Image of a BST.

Now given a sequence of integer keys, you are supposed to tell if it is the preorder traversal sequence of a BST or the mirror image of a BST.

Input Specification:

Each input file contains one test case. For each case, the first line contains a positive integer $N (\leqslant 1000)$. Then $N$ integer keys are given in the next line. All the numbers in a line are separated by a space.

Output Specification:

For each test case, first print in a line YES if the sequence is the preorder traversal sequence of a BST or the mirror image of a BST, or NO if not. Then if the answer is YES, print in the next line the postorder traversal sequence of that tree. All the numbers in a line must be separated by a space, and there must be no extra space at the end of the line.

Sample Input 1:

7
8 6 5 7 10 8 11

Sample Output 1:

YES
5 7 6 8 11 10 8

Sample Input 2:

7
8 10 11 8 6 7 5

Sample Output 2:

YES
11 8 10 7 5 6 8

Sample Input 3:

7
8 6 8 5 10 9 11

Sample Output 3:

NO

题目要点

本题 25 分，考察的是二叉搜索树的影用，难度较大，对树的理解有极大帮助。

注意，使用 Python 做树相关的题目时，可能会出现“非零返回”的错误提示。这可能是因为 Python 解释器的默认递归深度有限，测试样例的数据超过了默认的1000，手动设置递归深度为一个较大的数即可。

import sys
sys.setrecursionlimit(5000)

AC 源代码

import sys
sys.setrecursionlimit(5000)

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

class BST():
    def insert(self, root, val):
        if root == None:
            root = BTNode(val)
        elif val < root.val:
            root.left = self.insert(root.left, val)
        else:
            root.right = self.insert(root.right, val)
        return root

    def pre_order(self, root, lst):
        if root == None: return
        lst.append(root.val)
        self.pre_order(root.left, lst)
        self.pre_order(root.right, lst)

    def post_order(self, root, lst):
        if root == None: return
        self.post_order(root.left, lst)
        self.post_order(root.right, lst)
        lst.append(root.val)

    def pre_order_mirror(self, root, lst):
        if root == None: return
        lst.append(root.val)
        self.pre_order_mirror(root.right, lst)
        self.pre_order_mirror(root.left, lst)

    def post_order_mirror(self, root, lst):
        if root == None: return
        self.post_order_mirror(root.right, lst)
        self.post_order_mirror(root.left, lst)
        lst.append(root.val)

count = int(input())
seq = list(map(int, input().split()))

bt = BST()
tree = None
pre_seq = []
pre_seq_mirror = []
post_seq = []

for val in seq:
    tree = bt.insert(tree, val)

bt.pre_order(tree, pre_seq)
bt.pre_order_mirror(tree, pre_seq_mirror)

if seq == pre_seq:
    bt.post_order(tree, post_seq)
    print('YES')
    print(' '.join(map(str, post_seq)))
elif seq == pre_seq_mirror:
    bt.post_order_mirror(tree, post_seq)
    print('YES')
    print(' '.join(map(str, post_seq)))
else:
    print('NO')

参考来源

https://qsctech-sange.github.io/1043-Is-It-a-Binary-Search-Tree.html#%E4%BB%A3%E7%A0%81

https://docs.python.org/zh-cn/3.8/library/sys.html#sys.setrecursionlimit

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