There is a kind of balanced binary search tree named red-black tree in the data structure. It has the following 5 properties:

  • (1) Every node is either red or black.
  • (2) The root is black.
  • (3) Every leaf (NULL) is black.
  • (4) If a node is red, then both its children are black.
  • (5) For each node, all simple paths from the node to descendant leaves contain the same number of black nodes.

For example, the tree in Figure 1 is a red-black tree, while the ones in Figure 2 and 3 are not.

Figure 1 Figure 2 Figure 3

For each given binary search tree, you are supposed to tell if it is a legal red-black tree.

Input Specification:

Each input file contains several test cases. The first line gives a positive integer K (≤30) which is the total number of cases. For each case, the first line gives a positive integer N (≤30), the total number of nodes in the binary tree. The second line gives the preorder traversal sequence of the tree. While all the keys in a tree are positive integers, we use negative signs to represent red nodes. All the numbers in a line are separated by a space. The sample input cases correspond to the trees shown in Figure 1, 2 and 3.

Output Specification:

For each test case, print in a line "Yes" if the given tree is a red-black tree, or "No" if not.

Sample Input:

3
9
7 -2 1 5 -4 -11 8 14 -15
9
11 -2 1 -7 5 -4 8 14 -15
8
10 -7 5 -6 8 15 -11 17

Sample Output:

Yes
No
No
要求题目中写的很清楚,红黑树是二叉搜索树,所以给出前序遍历,那么中序遍历也可以知道(从小到大排序就是中序遍历),但负号不是代表大小,所以排序前,要取绝对值,然后建树,进行判断,按照题目要求,根结点必须是黑的(正的),
红色的儿子必须都是黑色,从某个点到所有的子孙叶子结点的路径包含黑色点个数相同。
代码:
#include <iostream>
#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std; struct tree {
int data;
tree *left,*right;
};
int pre[],in[];
int k,n,flag;
tree *build(int pre_l,int pre_r,int in_l,int in_r) {
tree *t = new tree();
t -> left = t -> right = NULL;
for(int i = in_l;i <= in_r;i ++) {
if(in[i] == abs(pre[pre_l])) {
if(i != in_l)t -> left = build(pre_l + ,pre_l + i - in_l,in_l,i - );
if(i != in_r)t -> right = build(pre_l + i - in_l + ,pre_r,i + ,in_r);
break;
}
}
t -> data = pre[pre_l];
return t;
}
int check(tree *t) {
if(t == NULL)return ;
if(t -> data < && (t -> left && t -> left -> data < || t -> right && t -> right -> data < )) {
flag = ;
return ;
}
int d = check(t -> left),e = check(t -> right);
if(d != e)flag = ;
return d + (t -> data > );///如果颜色为黑色,返回值加1
}
void drop(tree *t) {
if(t == NULL)return;
drop(t -> left);
drop(t -> right);
delete t;
}
int main() {
scanf("%d",&k);
while(k --) {
scanf("%d",&n);
for(int i = ;i < n;i ++) {
scanf("%d",&pre[i]);
in[i] = abs(pre[i]);///取绝对值
}
flag = ;
sort(in,in + n);
tree *head = build(,n - ,,n - );///建树
if(head -> data < )flag = ;///根结点不是黑色
else check(head);///检查是否满足
drop(head);///释放空间
puts(flag ? "Yes" : "No");
}
}

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