Pairs of Paths
Description
You are given a tree consisting of $n$ vertices, and $m$ simple vertex paths. Your task is to find how many pairs of those paths intersect at exactly one vertex. More formally you have to find the number of pairs $(i, j)$ $(1 \leq i < j \leq m)$ such that $path_i$ and $path_j$ have exactly one vertex in common.
First line contains a single integer $n$ $(1 \leq n \leq 3 \cdot 10^5)$.
Next $n - 1$ lines describe the tree. Each line contains two integers $u$ and $v$ $(1 \leq u, v \leq n)$ describing an edge between vertices $u$ and $v$.
Next line contains a single integer $m$ $(1 \leq m \leq 3 \cdot 10^5)$.
Next $m$ lines describe paths. Each line describes a path by it's two endpoints $u$ and $v$ $(1 \leq u, v \leq n)$. The given path is all the vertices on the shortest path from $u$ to $v$ (including $u$ and $v$).
Output a single integer — the number of pairs of paths that intersect at exactly one vertex.
Input
First line contains a single integer $n$ $(1 \leq n \leq 3 \cdot 10^5)$.
Next $n - 1$ lines describe the tree. Each line contains two integers $u$ and $v$ $(1 \leq u, v \leq n)$ describing an edge between vertices $u$ and $v$.
Next line contains a single integer $m$ $(1 \leq m \leq 3 \cdot 10^5)$.
Next $m$ lines describe paths. Each line describes a path by it's two endpoints $u$ and $v$ $(1 \leq u, v \leq n)$. The given path is all the vertices on the shortest path from $u$ to $v$ (including $u$ and $v$).
Output
Output a single integer — the number of pairs of paths that intersect at exactly one vertex.
Samples
5
1 2
1 3
1 4
3 5
4
2 3
2 4
3 4
3 5
2
1
3
1 1
1 1
1 1
3
5
1 2
1 3
1 4
3 5
6
2 3
2 4
3 4
3 5
1 1
1 2
7
Note
The tree in the first example and paths look like this. Pairs $(1,4)$ and $(3,4)$ intersect at one vertex.
In the second example all three paths contain the same single vertex, so all pairs $(1, 2)$, $(1, 3)$ and $(2, 3)$ intersect at one vertex.
The third example is the same as the first example with two additional paths. Pairs $(1,4)$, $(1,5)$, $(2,5)$, $(3,4)$, $(3,5)$, $(3,6)$ and $(5,6)$ intersect at one vertex.