1064. Complete Binary Search Tree (30)

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

思路：一般想到的是先构建树，然后再遍历，因为是Complete Binary Tree (CBT)的缘故，它的节点排列有规律，就没有用构建树实现

大概思想：先排序；再找出此树中最大满二叉树以外的叶子节点；再层次遍历满二叉树（总有序数列的中间开始向外扩散）

AC才能考代码：

#include <iostream>
#include <algorithm>
#include <vector>
#include <math.h>
#include <queue>
using namespace std;
vector<int> leaf;   //除满二叉树以外的叶子节点
vector<int> arr;    //节点记录
int N;

int main()
{
    cin>>N;
    arr.resize(N);
    for(int i=0;i<N;i++){
        cin>>arr[i];
    }
    if(N==1){
        cout<<arr[0];
        return 0;
    }
    sort(arr.begin(),arr.end());
    double ex = floor(log(N)/log(2));   //分离出的满二叉树的高度
    int leafNum = N - pow(2,ex)+1;      //除满二叉树以外的叶子节点的个数
    int index = 0;
    for(int i=0;i<leafNum;i++){         //满二叉树以外的叶子节点相邻之间应该相差一个节点，在有序数组中
        leaf.push_back(arr[index]);
        index += 2;
    }
    index = 0;
    for(vector<int>::iterator it=arr.begin();it!=arr.end();it++){   //从arr中除去非满二叉树的节点
        if(index==leafNum){
            break;
        }
        if((*it)==leaf[index]){
            arr.erase(it);
            index++;
        }
    }
    //模拟满二叉树的层次遍历，需用到队列
    int mid = arr.size()/2;
    int loc = mid+1;
    queue<int> qu;
    qu.push(mid);
    int totalCount = 0;
    int door = N-leafNum;
    while(!qu.empty()){
        int queueSize = qu.size();
        for(int k=0;k<queueSize;k++){
            int temp = qu.front();
            qu.pop();
            cout<<arr[temp]<<" ";
            if(totalCount<door-1){
                qu.push(temp-loc/2);
                qu.push(temp+loc/2);
                totalCount += 2;
            }
        }
        loc /= 2;
    }
    for(int i=0;i<leaf.size();i++){
        i<leaf.size()-1?cout<<leaf[i]<<" ":cout<<leaf[i];
    }
    return 0;
}