UESTC1559 Data Structure Problem [最大堆，最小堆，二叉搜索树]

Data Structure Problem

Time Limit: 1000 ms Memory Limit: 65536 kB Solved: 214 Tried: 795

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Description

 

Data structure is a fundamental course of Computer Science, so that each contestant is highly likely to solve this data structure problem.
A Heap data structure is a binary tree with the following properties:
1. It is a complete binary tree; that is, each level of the tree is completely filled, except possibly the bottom level. At this level, it is filled from left to right.
2. It satisfies the heap-order property: The key stored in each node is greater than or equal to the keys stored in its children.
So such a heap is sometimes called a max-heap. (Alternatively, if the comparison is reversed, the smallest element is always in the root node, which results in a min-heap.)
A binary search tree (BST), which may sometimes also be called an ordered or sorted binary tree, is a node-based binary tree data structure which has the following properties:
1. The left subtree of a node contains only nodes with keys less than (greater than) the node's key.
2. The right subtree of a node contains only nodes with keys greater than (less than) the node's key.
3. Both the left and right subtrees must also be binary search trees.
Given a complete binary tree with N keys, your task is to determine the type of it.
Note that either a max-heap or a min-heap is acceptable, and it is also acceptable for both increasing ordered BST and decreasing ordered BST.

 

Input

 

The first line of the input is T (no more than 100), which stands for the number of test cases you need to solve.
For each test case, the first line contains an integer N (1 <= N <= 1000), indicating the number of keys in the binary tree. On the second line, a permutation of 1 to N is given. The key stored in root node is given by the first integer, and the 2ith and 2i+1th integers are keys in the left child and right child of the ith integer respectively.

 

Output

 

For every test case, you should output "Case #k: " first, where k indicates the case number and counts from 1. Then output the type of the binary tree:
“Neither” --- It is neither a Heap nor a BST.
“Both” --- It is both a Heap and a BST.
“Heap” --- It is only a Heap.
“BST” --- It is only a BST.

 

Sample Input

 

4
1
1
3
1 2 3
3
2 1 3
4
2 1 3 4

 

Sample Output

 

Case #1: Both
Case #2: Heap
Case #3: BST
Case #4: Neither

 

Source

 

Sichuan State Programming Contest 2011

数据结构：判断最大堆，最小堆，二叉搜索树

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二叉搜索树：

二叉查找树（Binary Search Tree），或者是一棵空树，或者是具有下列性质的二叉树： 若它的左子树不空，则左子树上所有结点的值均小于它的根结点的值； 若它的右子树不空，则右子树上所有结点的值均大于它的根结点的值； 它的左、右子树也分别为二叉排序树。

堆：

堆是一种特殊的树形数据结构，每个结点都有一个值。通常我们所说的堆的数据结构，是指二叉堆。堆的特点是根结点的值最小（或最大），且根结点的两个子树也是一个堆
堆分为大根堆,小根堆,大根堆就是树的根结点大于叶子结点.

完全二叉树：

完全二叉树(Complete Binary Tree) 

　　若设二叉树的深度为h，除第 h 层外，其它各

层 (1～h-1) 的结点数都达到最大个数，第 h 层所有的结点都连续集中在最左边，这就是完全二叉树。 

　　完全二叉树是由满二叉树而引出来的。对于深度为K的，有N个结点的二叉树，当且仅当其每一个结点都与深度为K的满二叉树中编号从1至n的结点一一对应时称之为完全二叉树。 

　　若一棵二叉树至多只有最下面的两层上的结点的度数可以小于2，并且最下层上的结点都集中在该层最左边的若干位置上，则此二叉树成为完全二叉树。