F - Distributing Integers

Score : $600$ points

Problem Statement

We have a tree with $N$ vertices numbered $1$ to $N$. The $i$-th edge in this tree connects Vertex $a_i$ and $b_i$. For each $k=1, ..., N$, solve the problem below:

• Consider writing a number on each vertex in the tree in the following manner:
  • First, write $1$ on Vertex $k$.
  • Then, for each of the numbers $2, ..., N$ in this order, write the number on the vertex chosen as follows:
    • Choose a vertex that still does not have a number written on it and is adjacent to a vertex with a number already written on it. If there are multiple such vertices, choose one of them at random.
• Find the number of ways in which we can write the numbers on the vertices, modulo $(10^9+7)$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq a_i,b_i \leq N$
• The given graph is a tree.

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Input

Input is given from Standard Input in the following format:

$N$

$a_1$ $b_1$

$:$

$a_{N-1}$ $b_{N-1}$

Output

For each $k=1, 2, ..., N$ in this order, print a line containing the answer to the problem.

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3
1 2
1 3

2
1
1

The graph in this input is as follows:

For $k=1$, there are two ways in which we can write the numbers on the vertices, as follows:

• Writing $1, 2, 3$ on Vertex $1, 2, 3$, respectively
• Writing $1, 3, 2$ on Vertex $1, 2, 3$, respectively

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2
1 2

1
1

The graph in this input is as follows:

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5
1 2
2 3
3 4
3 5

2
8
12
3
3

The graph in this input is as follows:

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8
1 2
2 3
3 4
3 5
3 6
6 7
6 8

40
280
840
120
120
504
72
72

The graph in this input is as follows: