1155 Heap Paths （30 分）

In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: if P is a parent node of C, then the key (the value) of P is either greater than or equal to (in a max heap) or less than or equal to (in a min heap) the key of C. A common implementation of a heap is the binary heap, in which the tree is a complete binary tree. (Quoted from Wikipedia at https://en.wikipedia.org/wiki...

One thing for sure is that all the keys along any path from the root to a leaf in a max/min heap must be in non-increasing/non-decreasing order.

Your job is to check every path in a given complete binary tree, in order to tell if it is a heap or not.

Input Specification:

Each input file contains one test case. For each case, the first line gives a positive integer N (1<N≤1,000), the number of keys in the tree. Then the next line contains N distinct integer keys (all in the range of int), which gives the level order traversal sequence of a complete binary tree.

Output Specification:

For each given tree, first print all the paths from the root to the leaves. Each path occupies a line, with all the numbers separated by a space, and no extra space at the beginning or the end of the line. The paths must be printed in the following order: for each node in the tree, all the paths in its right subtree must be printed before those in its left subtree.

Finally print in a line Max Heap if it is a max heap, or Min Heap for a min heap, or Not Heap if it is not a heap at all.

Sample Input 1:

8

98 72 86 60 65 12 23 50

Sample Output 1:

98 86 23

98 86 12

98 72 65

98 72 60 50

Max Heap

Sample Input 2:

8

8 38 25 58 52 82 70 60

Sample Output 2:

8 25 70

8 25 82

8 38 52

8 38 58 60

Min Heap

Sample Input 3:

8

10 28 15 12 34 9 8 56

Sample Output 3:

10 15 8

10 15 9

10 28 34

10 28 12 56

Not Heap

题目重点信息提取：1.输入： positive integer N 正整数， N distinct integer keys ，N个互不相等 的整数

给出一颗完全二叉树 **level order** traversal sequence of a **complete binary tree**
    提取重点翻译：level order 与 complete binary tree ，input一个完全二叉树的层序遍历序列
    2.输出： 重点翻译：first print **all the paths from the root to the leaves**
    先打印出所有从根结点到叶子结点的路径，all the paths in its **right subtree** must be printed **before** those in its **left subtree**，到右子树的路径要先于到左子树路径打印。
    画图对应样例的输入输出也可以快速判断出来

思路：深度遍历并打印出所有的路径（先右后左），用vector存储路径上的所有结点，通过push和pop回溯，维护路径，关于 index <= n ，由于是先右后左，需要对只有左叶子结点而无右叶子结点的点进行特判。

#include <iostream>
#include <stdio.h>
#include <vector>
using namespace std;

int n,a[1001],isMaxHeap = 1,isMinHeap = 1;
vector<int> v;
void R_dfs(int index){  //从右至左的深度优先遍历
    if(index * 2 > n && index * 2 + 1 > n){
        if( index <= n){  //由于是先右后左，需要对只有左叶子结点而无右叶子结点的点进行特判
            for(int i = 0; i < v.size(); i++)
                printf("%d%s",v[i], i != v.size()-1 ? " " : "\n");
        }
    }
    else{
        v.push_back(a[index *2 + 1]); //深度遍历右子树
        R_dfs(index * 2 + 1);
        v.pop_back();
        v.push_back(a[index * 2]); //深度优先遍历左子树
        R_dfs(index * 2);
        v.pop_back();
    }
}
int main()
{
    scanf("%d",&n);
    for(int i = 1; i <= n; i++){ //这里从i=1开始，方便后续对二叉树有无右子树进行判断
        scanf("%d",&a[i]);
    }
    v.push_back(a[1]);
    R_dfs(1);
    for(int i = 2; i <= n; i++){ //判断大小顶堆
        if(a[i/2] > a[i])
            isMinHeap = 0;
        else if(a[i/2] < a[i])
            isMaxHeap = 0;
    }
    if(isMinHeap == 1)
        printf("%s\n","Min Heap");
    else
        printf("%s\n",isMaxHeap == 1 ? "Max Heap" : "Not Heap");
    return 0;
}