CF1260F.Colored Tree

You're given a tree with $n$ vertices. The color of the $i$ -th vertex is $h_{i}$ .

The value of the tree is defined as $\sum\limits_{h_{i} = h_{j}, 1 \le i < j \le n}{dis(i,j)}$ , where $dis(i,j)$ is the number of edges on the shortest path between $i$ and $j$ .

The color of each vertex is lost, you only remember that $h_{i}$ can be any integer from $[l_{i}, r_{i}]$ (inclusive). You want to calculate the sum of values of all trees meeting these conditions modulo $10^9 + 7$ (the set of edges is fixed, but each color is unknown, so there are $\prod\limits_{i = 1}^{n} (r_{i} - l_{i} + 1)$ different trees).

The first line contains one integer $n$ ( $2 \le n \le 10^5$ ) — the number of vertices.

Then $n$ lines follow, each line contains two integers $l_i$ and $r_i$ ( $1 \le l_i \le r_i \le 10^5$ ) denoting the range of possible colors of vertex $i$ .

Then $n - 1$ lines follow, each containing two integers $u$ and $v$ ( $1 \le u, v \le n$ , $u \ne v$ ) denoting an edge of the tree. It is guaranteed that these edges form a tree.

Print one integer — the sum of values of all possible trees, taken modulo $10^9 + 7$ .

In the first example there are four different ways to color the tree (so, there are four different trees):

• a tree with vertices colored as follows: $\lbrace 1,1,1,1 \rbrace$ . The value of this tree is $dis(1,2)+dis(1,3)+dis(1,4)+dis(2,3)+dis(2,4)+dis(3,4) = 10$ ;
• a tree with vertices colored as follows: $\lbrace 1,2,1,1 \rbrace$ . The value of this tree is $dis(1,3)+dis(1,4)+dis(3,4)=4$ ;
• a tree with vertices colored as follows: $\lbrace 1,1,1,2 \rbrace$ . The value of this tree is $dis(1,2)+dis(1,3)+dis(2,3)=4$ ;
• a tree with vertices colored as follows: $\lbrace 1,2,1,2 \rbrace$ . The value of this tree is $dis(1,3)+dis(2,4)=4$ .

Overall the sum of all values is $10+4+4+4=22$ .