103126: [Atcoder]ABC312 G - Avoid Straight Line

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