1064 Complete Binary Search Tree (30 分)
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:已知树是一棵完全二叉搜索树:即如果该层不是最后一层,则无叶子节点;最后一层没有只有右叶子无左叶子的情况。给定一个序列,输出该树的层序。
解题思路:
- BST的性质:中序一定是有序序列->sort即可;
- CBT的性质:节点编号满足父节点n,左孩子2*n,右孩子2*n+1(此时n=1)的规律;
- 用dfs遍历中序序列,再按照CBT的下标保存就可以得到层序。
#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;
#define rep(i,j,k) for(int i=j;i<k;i++)
const int maxn=1100;
int in[maxn],level[maxn];
int n,index=0;
void dfs(int u){
if(u>n) return;
dfs(2*u);
level[u] = in[index++];
dfs(2*u+1);
}
int main(){
std::ios::sync_with_stdio(false);
cin>>n;
rep(i,0,n)
cin>>in[i];
sort(in,in+n);
dfs(1);
rep(i,1,n+1)
printf("%d%s",level[i],i==n?"":" ");
return 0;
}