A12367 | Edgy Trees

You are given a tree (a connected undirected graph without cycles) of $n$ vertices. Each of the $n - 1$ edges of the tree is colored in either black or red.

You are also given an integer $k$ . Consider sequences of $k$ vertices. Let's call a sequence $[a_1, a_2, \ldots, a_k]$ good if it satisfies the following criterion:

- We will walk a path (possibly visiting same edge/vertex multiple times) on the tree, starting from $a_1$ and ending at $a_k$ .
- Start at $a_1$ , then go to $a_2$ using the shortest path between $a_1$ and $a_2$ , then go to $a_3$ in a similar way, and so on, until you travel the shortest path between $a_{k-1}$ and $a_k$ .
- If you walked over at least one black edge during this process, then the sequence is good.

Consider the tree on the picture. If $k=3$ then the following sequences are good: $[1, 4, 7]$ , $[5, 5, 3]$ and $[2, 3, 7]$ . The following sequences are not good: $[1, 4, 6]$ , $[5, 5, 5]$ , $[3, 7, 3]$ .

There are $n^k$ sequences of vertices, count how many of them are good. Since this number can be quite large, print it modulo $10^9+7$ .

The first line contains two integers $n$ and $k$ ( $2 \le n \le 10^5$ , $2 \le k \le 100$ ), the size of the tree and the length of the vertex sequence.

Each of the next $n - 1$ lines contains three integers $u_i$ , $v_i$ and $x_i$ ( $1 \le u_i, v_i \le n$ , $x_i \in \{0, 1\}$ ), where $u_i$ and $v_i$ denote the endpoints of the corresponding edge and $x_i$ is the color of this edge ( $0$ denotes red edge and $1$ denotes black edge).

Print the number of good sequences modulo $10^9 + 7$ .