PAT甲级2018冬季7-4 1155 Heap Paths（30 分）

雨点打透心脏的1/2处 2023-02-24 08:46 4阅读 0赞

[算法笔记总目录][Link 1]  
[关键英语单词解释][Link 2]  
[1155 Heap Paths（30 分）][1155 Heap Paths_30]  
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/Heap\_(data\_structure))

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

[代码一来自柳诺][Link 3]  
题目大意：给出一颗完全二叉树，打印出从根节点到所有叶节点的路径，打印顺序先右后左，即先序遍历的镜像。然后判断该树是大顶堆、小顶堆或者不是堆～

分析：1.深搜打印出所有路径（从右往左，即先序的镜像），vector保存一路上的节点，通过push和pop回溯，维护路径，index <= n是对只有左叶节点没有右叶节点的点特判  
2.判断是否为堆：从第二个节点开始遍历，如果比父节点小，就不是小顶堆，如果比父节点大，就不是大顶堆～

#include <iostream>
    #include <vector>
    using namespace std;
    vector<int> v;
    int a[1009], n, isMin = 1, isMax = 1;
    void dfs(int index) { 
        if (index * 2 > n && index * 2 + 1 > n) { 
            if (index <= n) { 
                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]);
            dfs(index * 2 + 1);
            v.pop_back();
            v.push_back(a[index * 2]);
            dfs(index * 2);
            v.pop_back();
        }
    }
    int main() { 
        cin >> n;
        for (int i = 1; i <= n; i++)
            scanf("%d", &a[i]);
        v.push_back(a[1]);
        dfs(1);
        for (int i = 2; i <= n; i++) { 
            if (a[i/2] > a[i]) isMin = 0;
            if (a[i/2] < a[i]) isMax = 0;
        }
        if (isMin == 1)
            printf("Min Heap");
        else 
            printf("%s", isMax == 1 ? "Max Heap" : "Not Heap"); 
        return 0;
    }

**代码二**  
完全层序建树（练习使用，考试第一种方法即可）

#include<bits/stdc++.h>
    using namespace std;
    struct node{ 
    	int val;
    	node *lchild,*rchild;
    };
    int n,level[1010],x=0,a[1010],isMin = 1, isMax = 1;
    node *create(int x,node *root)//1号位开始 
    { 
    	if(x>n) return NULL;
    	root->val=level[x];
        root->lchild=create(x*2,new node);
        root->rchild=create(x*2+1,new node);
    	return root;
    }
    
    void printTree(node *root){ 
    	 if(root->rchild== NULL && root->lchild== NULL){ 
    	 	printf("%d",a[0]); 
    	    for(int i=1;i<x;i++) printf(" %d",a[i]); 
    	    printf(" %d\n",root->val);
    	 	return;
    	 }
    	 a[x]=root->val;
    	 x++;
    	if(root->rchild != NULL)printTree(root->rchild);
    	if(root->lchild != NULL)printTree(root->lchild);
    	x--;
    }
    int main(){ 
    	scanf("%d",&n);
    	node *root = new node;
    	for (int i = 1; i <= n; i++)
            scanf("%d", &level[i]);
        printTree(create(1,root));
         for (int i = 2; i <= n; i++) { 
            if (level[i/2] > level[i]) isMin = 0;
            if (level[i/2] < level[i]) isMax = 0;
        }
        if (isMin == 1)
            printf("Min Heap");
        else 
            printf("%s", isMax == 1 ? "Max Heap" : "Not Heap");
        return 0;
    }

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzIxNDgwNjA3_size_16_color_FFFFFF_t_70]

[Link 1]: https://blog.csdn.net/qq_21480607/article/details/106660085
[Link 2]: https://blog.csdn.net/qq_21480607/article/details/106987591
[1155 Heap Paths_30]: https://pintia.cn/problem-sets/994805342720868352/problems/1071785408849047552
[Link 3]: https://blog.csdn.net/liuchuo/article/details/84973009
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzIxNDgwNjA3_size_16_color_FFFFFF_t_70]: /images/20230209/3d250ffd383c4f369042f25d26098313.png