F - Tree and Constraints

Score : $600$ points

Problem Statement

We have a tree with $N$ vertices numbered $1$ to $N$. The $i$-th edge in this tree connects Vertex $a_i$ and Vertex $b_i$.
Consider painting each of these edges white or black. There are $2^{N-1}$ such ways to paint the edges. Among them, how many satisfy all of the following $M$ restrictions?

• The $i$-th $(1 \leq i \leq M)$ restriction is represented by two integers $u_i$ and $v_i$, which mean that the path connecting Vertex $u_i$ and Vertex $v_i$ must contain at least one edge painted black.

Constraints

• $2 \leq N \leq 50$
• $1 \leq a_i,b_i \leq N$
• The graph given in input is a tree.
• $1 \leq M \leq \min(20,\frac{N(N-1)}{2})$
• $1 \leq u_i < v_i \leq N$
• If $i \not= j$, either $u_i \not=u_j$ or $v_i\not=v_j$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$a_1$ $b_1$

$:$

$a_{N-1}$ $b_{N-1}$

$M$

$u_1$ $v_1$

$:$

$u_M$ $v_M$

Output

Print the number of ways to paint the edges that satisfy all of the $M$ conditions.

3
1 2
2 3
1
1 3

3

The tree in this input is shown below:



All of the $M$ restrictions will be satisfied if Edge $1$ and $2$ are respectively painted (white, black), (black, white), or (black, black), so the answer is $3$.

2
1 2
1
1 2

1

The tree in this input is shown below:



All of the $M$ restrictions will be satisfied only if Edge $1$ is painted black, so the answer is $1$.

5
1 2
3 2
3 4
5 3
3
1 3
2 4
2 5

9

The tree in this input is shown below:

8
1 2
2 3
4 3
2 5
6 3
6 7
8 6
5
2 7
3 5
1 6
2 8
7 8

62

The tree in this input is shown below: