PAT 1043. Is It a Binary Search Tree (25)

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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 (<=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

给出的是前序序列，序列中的第一个点是树中的根。从根的下一个结点开始遍历，直到找到一个比根还要大的结点；再从树的尾部往前找到第一个比根小的结点。如果这两个结点的下标不是相差一的关系，就说明这个树不满足二叉搜索树的定义。同理，可以得到镜像的二叉树是否满足关系。

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#include <cstdio>
#include <vector>

int *pre;

void getPost(std::vector<int> &post, int s, int e, bool mirror) {
    if (s > e) {
        return;
    }
    int i = s + 1, j = e;
    if (!mirror) {
        while (i <= e && pre[s] > pre[i]) i++;
        while (j > s && pre[s] <= pre[j]) j--;
    } else {
        while (i <= e && pre[s] <= pre[i]) i++;
        while (j > s && pre[s] > pre[j]) j--;
    }
    if (i - j != 1) return;
    getPost(post, s + 1, j, mirror);
    getPost(post, i, e, mirror);
    post.push_back(pre[s]);
}

int main() {
    int n = 0;
    scanf("%d", &n);
    pre = new int[n];
    for (int i = 0; i < n; i++) {
        scanf("%d", &pre[i]);
    }
    std::vector<int> post;
    getPost(post, 0, n - 1, false);
    if (post.size() != n) {
        post.clear();
        getPost(post, 0, n - 1, true);
    }
    if (post.size() == n) {
        printf("YES\n%d", post[0]);
        for (int i = 1; i < post.size(); i++) {
            printf(" %d", post[i]);
        }
    } else {
        printf("NO");
    }
    delete[] pre;
    return 0;
}