PAT-1064 Complete Binary Search Tree（完全二叉树）

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

题目大意：给出一个构成完全二叉树（除了最后一层的右边可能缺少部分节点，其余层均达到最大节点数）的序列，要求输出该完全二叉树的层次遍历。

主要思想：根据完全二叉树的特性可以用一个数组很方便的表示出来，将根节点的索引设为1，之后每个索引为i的节点，其左节点为2i，右节点为2i+1，然后对于输入的序列进行排序后按照中序遍历依次填入二叉树数组，这样完全二叉树就构造成功了，而最后一步的层次遍历输出其实就是对数组的顺序输出。

#include <cstdio>
#include <algorithm>
int n;						//节点个数
int index = 0;				//序列数组的索引
int a[1005];				//输入的序列数组
int node[1005];				//完全二叉树数组
using namespace std;

void travel(int i) {
    if (i > n)  return;
    travel(2*i);
    node[i] = a[index++];
    travel(2*i+1);
}

int main(void) {
    int i;

    scanf("%d", &n);
    for (i = 0; i < n; i++) {
        scanf("%d", &a[i]);
    }
    sort(a, a+n);
    travel(1);				//中序遍历构造完全二叉树
	//层次遍历输出
    for (i = 1; i < n; i++)
        printf("%d ", node[i]);
    printf("%d\n", node[i]);

    return 0;
}