Problem:
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.

Sample Input 1

3
1 2
2 3
1
1 3

Sample Output 1

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.

Sample Input 2

2
1 2
1
1 2

Sample Output 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.

Sample Input 3

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

Sample Output 3

9

The tree in this input is shown below:

Sample Input 4

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

Sample Output 4

62

The tree in this input is shown below:

Solution in PHP (7.4.4) (judge verdict: Time Limit Exceeded)

<?php

fscanf(STDIN, "%d", $n);

$vertices = [];
$edges = [];
for ($i = 0; $i < $n - 1; ++$i) {
    fscanf(STDIN, "%d%d", $a, $b);

    if (!isset($vertices[$a])) {
        $vertices[$a] = new Vertex($a);
        $edges[$a] = [];
    }

    if (!isset($vertices[$b])) {
        $vertices[$b] = new Vertex($b);
        $edges[$b] = [];
    }

    $edges[$a][$b] = new Edge($i, $vertices[$a], $vertices[$b]);
    $edges[$b][$a] = new Edge($i, $vertices[$b], $vertices[$a]);
}

$tree = new Tree($vertices, $edges);

fscanf(STDIN, "%d", $m);

$paths = [];
for ($i = 0; $i < $m; ++$i) {
    fscanf(STDIN, "%d%d", $u, $v);

    $lca = $tree->findLCA($u, $v);
    $path_u = $tree->getPathsIdList($lca, $u);
    $path_v = $tree->getPathsIdList($lca, $v);

    $paths[] = array_merge($path_u, $path_v);
}

$answer = 0;
for ($i = 0; $i < 2 ** $m; ++$i) {
    $restriction = [];
    $will_add = true;
    for ($j = 0; $j < $m; ++$j) {
        if ($i & 2 ** $j) {
            $restriction = array_merge($restriction, $paths[$j]);
            $will_add = !$will_add;
        }
    }

    $restriction_count = 2 ** ($n - 1 - count(array_unique($restriction)));
    if ($will_add) {
        $answer += $restriction_count;
    } else {
        $answer -= $restriction_count;
    }
}

echo $answer;

class Tree
{
    public $vertices;

    public function __construct(array $vertices, array $edges)
    {
        $this->vertices = $vertices;

        $this->vertices[1]->setDepth(0);

        $this->composeTree($this->vertices[1], $edges);
    }

    protected function composeTree(Vertex $vertex, array $edges)
    {
        $this->edge_from_root[$vertex->id] = [];
        foreach ($edges[$vertex->id] as $child_id => $edge) {
            $vertex->addChild($this->vertices[$child_id], $edge);

            unset($edges[$child_id][$vertex->id]);

            $this->composeTree($this->vertices[$child_id], $edges);
        }
    }

    public function findLCA(int $i, int $j): int
    {
        $vertex_i = $this->vertices[$i];
        $vertex_j = $this->vertices[$j];

        $i_parent = $vertex_i->depth > $vertex_j->depth ? $vertex_i->findNthParent($vertex_i->depth - $vertex_j->depth) : $vertex_i;
        $j_parent = $vertex_i->depth > $vertex_j->depth ? $vertex_j : $vertex_j->findNthParent($vertex_j->depth - $vertex_i->depth);
        while ($i_parent !== $j_parent) {
            $i_parent = $i_parent->findNthParent(1);
            $j_parent = $j_parent->findNthParent(1);
        }

        return $i_parent->id;
    }

    public function getPathsIdList(int $source_id, int $target_id): array
    {
        $edge_id_list_from_root_to_source = $this->vertices[$source_id]->edge_id_list_from_root;
        $edge_id_list_from_root_to_target = $this->vertices[$target_id]->edge_id_list_from_root;

        return array_diff($edge_id_list_from_root_to_target, $edge_id_list_from_root_to_source);
    }
}

class Edge
{
    public $id;
    public $source;
    public $target;

    public function __construct(int $id, Vertex $source, Vertex $target)
    {
        $this->id = $id;
        $this->source = $source;
        $this->target = $target;
    }
}

class Vertex
{
    public $id;
    public $parent;
    public $children = [];
    public $edge_id_list_from_root = [];
    public $depth;

    public function __construct(int $id)
    {
        $this->id = $id;
    }

    public function addChild(Vertex $child, Edge $edge_to_child): void
    {
        $this->children[] = $child;

        $edge_id_list_from_root_to_child = $this->edge_id_list_from_root;
        $edge_id_list_from_root_to_child[] = $edge_to_child->id;

        $child->setParent($this);
        $child->setEdgeFromRoot($edge_id_list_from_root_to_child);
        $child->setDepth($this->depth + 1);
    }

    public function setParent(Vertex $parent): void
    {
        $this->parent = $parent;
    }

    public function setEdgeFromRoot(array $edge_id_list_from_root): void
    {
        $this->edge_id_list_from_root = $edge_id_list_from_root;
    }

    public function setDepth(int $depth): void
    {
        $this->depth = $depth;
    }

    public function findNthParent(int $generation): ?self
    {
        if ($generation === 0) {
            return $this;
        }

        if ($this->parent === null) {
            return null;
        }
        return $this->parent->findNthParent($generation - 1);
    }
}