CF1139C 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 $ .

N/A

N/A

In the first example, all sequences ( $ 4^4 $ ) of length $ 4 $ except the following are good: - $ [1, 1, 1, 1] $ - $ [2, 2, 2, 2] $ - $ [3, 3, 3, 3] $ - $ [4, 4, 4, 4] $ In the second example, all edges are red, hence there aren't any good sequences.