Score : $550$ points

问题描述

给定一棵包含 $N$ 个顶点的树。顶点编号从 $1$ 到 $N$，第 $i$ 条边连接顶点 $A_i$ 和顶点 $B_i$。

求满足以下条件的整数元组 $(i,j,k)$ 的数量：

• $1 \leq i < j < k \leq N$；
• 所给定的树中不存在一条简单路径，该路径包含了顶点 $i$、$j$ 和 $k$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a tree with $N$ vertices. The vertices are numbered from $1$ through $N$, and the $i$-th edge connects vertex $A_i$ and vertex $B_i$.
Find the number of tuples of integers $(i,j,k)$ such that:

• $1 \leq i < j < k \leq N$; and
• the given tree does not contain a simple path that contains all of vertices $i$, $j$, and $k$.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq A_i, B_i \leq N$
• The given graph is a tree.
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $B_1$

$\vdots$

$A_{N-1}$ $B_{N-1}$

Output

Print the answer.

Sample Input 1

5
1 2
2 3
2 4
1 5

Sample Output 1

2

Two tuples satisfy the conditions: $(i,j,k) = (1,3,4),(3,4,5)$.

Sample Input 2

6
1 2
2 3
3 4
4 5
5 6

Sample Output 2

0

Sample Input 3

12
1 6
3 4
10 4
5 9
3 1
2 3
7 2
2 12
1 5
6 8
4 11

Sample Output 3

91

update @ 2024/3/10 08:54:15