A1064. Complete Binary Search Tree

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.

A Complete Binary Tree (CBT) is a tree that is completely filled, with the possible exception of the bottom level, which is filled from left to right.

Now given a sequence of distinct non-negative integer keys, a unique BST can be constructed if it is required that the tree must also be a CBT. You are supposed to output the level order traversal sequence of this BST.

Input Specification:

Each input file contains one test case. For each case, the first line contains a positive integer N (<=1000). Then N distinct non-negative integer keys are given in the next line. All the numbers in a line are separated by a space and are no greater than 2000.

Output Specification:

For each test case, print in one line the level order traversal sequence of the corresponding complete binary search 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:

10
1 2 3 4 5 6 7 8 9 0

Sample Output:

6 3 8 1 5 7 9 0 2 4

#include<cstdio>
#include<iostream>
#include<vector>
#include<algorithm>
using namespace std;
int tree[10001], N, index = 0, num[10001];
bool cmp(int a, int b){
    return a < b;
}
void inOrder(int root){
    if(root > N)
        return;
    inOrder(root * 2);
    tree[root] = num[index++];
    inOrder(root * 2 + 1);
}
int main(){
    scanf("%d", &N);
    for(int i = 0; i < N; i++)
        scanf("%d", &num[i]);
    sort(num, num + N, cmp);
    inOrder(1);
    for(int i = 1; i <= N; i++){
        if(i != N)
            printf("%d ", tree[i]);
        else printf("%d", tree[i]);
    }
    cin >> N;
    return 0;
}

总结：

1、题意：给出一组数字，要求将它们建立成一颗二叉搜索树。因为结果不唯一，所以加了限制条件：要求搜索树是一颗完全二叉树。

2、二叉搜索树的中序序列是从小到大的有序数列，所以对初始序列排序后就能得到搜索树的中序序列。

3、完全二叉树的树形状在给出节点个数N之后就是已知的。所以相当于已经知道了答案所求树的形状，但仅仅是树的节点没有填入值罢了。由于搜索树的中序序列已知，只需要按照中序遍历完全二叉树，在遍历的过程中填入搜索树的中序序列值即可。