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Saturday 16 July 2022

[Solution] Partial Virtual Trees Codeforces Solution



F. Partial Virtual Trees
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Kawashiro Nitori is a girl who loves competitive programming. One day she found a rooted tree consisting of n vertices. The root is vertex 1. As an advanced problem setter, she quickly thought of a problem.

Kawashiro Nitori has a vertex set U={1,2,,n}. She's going to play a game with the tree and the set. In each operation, she will choose a vertex set T, where T is a partial virtual tree of U, and change U into T.

A vertex set S1 is a partial virtual tree of a vertex set S2, if S1 is a subset of S2S1S2, and for all pairs of vertices i and j in S1LCA(i,j) is in S1, where LCA(x,y) denotes the lowest common ancestor of vertices x and y on the tree. Note that a vertex set can have many different partial virtual trees.

Kawashiro Nitori wants to know for each possible k, if she performs the operation exactly k times, in how

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 many ways she can make U={1} in the end? Two ways are considered different if there exists an integer z (1zk) such that after z operations the sets U are different.

Since the answer could be very large, you need to find it modulo p. It's guaranteed that p is a prime number.

Input

The first line contains two integers n and p (2n2000108+7p109+9). It's guaranteed that p is a prime number.

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Each of the next n1 lines contains two integers uivi (1ui,vin), representing an edge between ui and vi.

It is guaranteed that the given edges form a tree.

Output

The only line contains n1 integers — the answer modulo p for k=1,2,,n1.

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