D. Forest Game

Codeforces

IDCF10123D

Time4000ms

Memory512MB

Difficulty—

English · Original

Formal · Original

Consider the following boring game about removing nodes from a forest. Initially, the forest contains only one tree with N nodes, and your initial score is 0. The game then goes as follows: Please calculate the expected value of your final score multiplied by N!, modulo 109 + 7. The first line of input contains one integer N indicating the number of nodes in the initial tree. Each of the following N - 1 lines contains two integers x and y, indicating that x-th node and y-th node are connected by an edge in the given tree. The nodes are numbered from 1 to N. Output one number: the expected value of the final score of this boring game multiplied by N!, modulo 109 + 7. ## Input The first line of input contains one integer N indicating the number of nodes in the initial tree.Each of the following N - 1 lines contains two integers x and y, indicating that x-th node and y-th node are connected by an edge in the given tree. The nodes are numbered from 1 to N. 1 ≤ N ≤ 105 1 ≤ x, y ≤ N the given graph is a tree ## Output Output one number: the expected value of the final score of this boring game multiplied by N!, modulo 109 + 7. [samples]

**Definitions** Let $ N \in \mathbb{Z}^+ $ be the number of nodes in the initial tree. Let $ T = (V, E) $ be a tree with $ V = \{1, 2, \dots, N\} $ and $ |E| = N - 1 $. Let $ \sigma \in S_N $ be a uniformly random permutation of the nodes (representing a random removal order). Let $ S(\sigma) $ be the final score after removing nodes in order $ \sigma $, where at each step, removing a node contributes $ 1 $ to the score if and only if it is a leaf at the time of removal. **Constraints** - $ 1 \leq N \leq 10^5 $ - The graph is a tree. **Objective** Compute: $$ \mathbb{E}[S(\sigma)] \cdot N! \mod (10^9 + 7) $$ which is equivalent to: $$ \sum_{\sigma \in S_N} S(\sigma) \mod (10^9 + 7) $$

API Response (JSON)

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