P4434 [COCI 2017/2018 #2] ​​Usmjeri

We are given a tree with N nodes denoted with different positive integers from 1 to N. Additionally, you are given M node pairs from the tree in the form of ($a_1$ , $b_1$ ), ($a_2$ , $b_2$ ), …, ($a_M$ , $b_M$ ). We need to direct each edge of the tree so that for each given node pair ($a_i$ , $b_i$ ) there is a path from $a_i$ to $b_i$ or from $b_i$ to $a_i$ . How many different ways are there to achieve this? Since the solution can be quite large, determine it modulo $10^{9}+7$.

The first line of input contains the positive integers N and M (1 ≤ N, M ≤ $3*10^5$), the number of nodes in the tree and the number of given node pairs, respectively. Each of the following N - 1 lines contains two positive integers, the labels of the nodes connected with an edge. The $i_{th}$ of the following M lines contains two different positive integers $a_{i}$ and $b_{i}$ , the labels of the nodes from the $i_{th}$ node pair. All node pairs will be mutually different.

You must output a single line containing the total number of different ways to direct the edges of the tree that meet the requirement from the task, modulo $10^{9}+7$.

In test cases worth 20% of total points, the given tree will be a chain. In other words, node i will be connected with an edge to node i + 1 for all i < N. In additional test cases worth 40% of total points, it will hold N, M ≤ $5*10^3$. A tree is a graph that consists of N nodes and N - 1 edges such that there exists a path from each node to each other node.