Problem:
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.

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.

Sample Input 1

3
1 2
1 3

Sample Output 1

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

Sample Input 2

2
1 2

Sample Output 2

1
1

The graph in this input is as follows:

Sample Input 3

5
1 2
2 3
3 4
3 5

Sample Output 3

2
8
12
3
3

The graph in this input is as follows:

Sample Input 4

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

Sample Output 4

40
280
840
120
120
504
72
72

The graph in this input is as follows:

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import javax.management.loading.MLet;
import java.io.*;
import java.util.*;
import java.lang.*;

class Main {
	static final int M = 1000000007;

	private static List<Integer>[] g;
	private static int n;
	private static long[] dp;
	private static long[] dp2;
	private static Combination c;
	private static int[] size;

	private static void dfs1(int v, int p) {
		dp[v] = 1;
		for (int next : g[v]) {
			if (next == p) continue;
			dfs1(next, v);
			size[v] += size[next];
			dp[v] = dp[v] * dp[next] % M * c.choose(size[v] - 1, size[next]) % M;
		}
	}

	private static void dfs2(int v, int p) {
		for (int next : g[v]) {
			if (next == p) continue;
			//long a = dp2[p] * pow(dp[next], M - 2, M) * M
			//		* pow(c.choose(n - 1, size[next]), M - 2, M) % M;
			dp2[next] = dp2[v] * pow(c.choose(n - 1, size[next]), M - 2, M) % M * c.choose(n - 1, size[next] - 1) % M;
			dfs2(next, v);
		}
	}

	private static long pow(long x, long n, long M) {
		long a = 1;
		while (n > 0) {
			if ((n & 1) == 1) a = (a * x) % M;
			x = (x * x) % M;
			n >>= 1;
		}
		return a;
	}

	public static void main(String[] args) {
		final FastScanner sc = new FastScanner(System.in);
		n = sc.nextInt();
		g = new List[n];
		c = new Combination(n + 1, M);

		dp = new long[n];
		dp2 = new long[n];
		size = new int[n];
		Arrays.fill(size, 1);

		for (int i = 0; i < n; i++) {
			g[i] = new ArrayList<>();
		}

		for (int i = 0; i < n - 1; i++) {
			int a = sc.nextInt() - 1;
			int b = sc.nextInt() - 1;
			g[a].add(b);
			g[b].add(a);
		}

		dfs1(0, -1);
		dp2[0] = dp[0];
		dfs2(0, -1);

		StringBuilder sb = new StringBuilder();
		for (int i = 0; i < n; i++) {
			sb.append(dp2[i]).append("\n");
		}

		System.out.print(sb);
	}

	private static class Combination {
		private final int M;
		private long[] fact;
		private long[] ifact;
		private long[] inv;

		public Combination(int n, int M) {
			this.M = M;
			fact = new long[n + 1];
			ifact = new long[n + 1];
			inv = new long[n + 1];

			fact[0] = fact[1] = 1;
			ifact[0] = ifact[1] = 1;
			inv[1] = 1;

			for (int i = 2; i <= n; i++) {
				fact[i] = fact[i - 1] * i % M;
				// p = (p / a * a) + (p % a)
				// 0 ≡ (p / a * a) + (p % a)          (mod p)
				// a^(-1) ≡ - (p % a)^(-1) * (p / a)  (mod p)
				inv[i] = M - inv[M % i] * (M / i) % M;
				ifact[i] = ifact[i - 1] * inv[i] % M;
			}
		}

		public long choose(int n, int k) {
			if (n < k) return 0;
			if (n < 0 || k < 0) return 0;
			return fact[n] * (ifact[k] * ifact[n - k] % M) % M;
		}

		public long inv(int a) {
			return inv[a];
		}

		public long fact(int a) {
			return fact[a];
		}
	}

	static class FastScanner {
		private final InputStream in;
		private final byte[] buffer = new byte[1024];
		private int ptr = 0;
		private int buflen = 0;

		public FastScanner(InputStream source) {
			in = source;
		}

		private boolean hasNextByte() {
			if (ptr < buflen) {
				return true;
			} else {
				ptr = 0;
				try {
					buflen = in.read(buffer);
				} catch (IOException e) {
					e.printStackTrace();
				}
				if (buflen <= 0) {
					return false;
				}
			}
			return true;
		}

		private int readByte() {
			if (hasNextByte()) return buffer[ptr++];
			else return -1;
		}

		private static boolean isPrintableChar(int c) {
			return 33 <= c && c <= 126;
		}

		public boolean hasNext() {
			while (hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;
			return hasNextByte();
		}

		public String next() {
			if (!hasNext()) throw new NoSuchElementException();
			StringBuilder sb = new StringBuilder();
			int b = readByte();
			while (isPrintableChar(b)) {
				sb.appendCodePoint(b);
				b = readByte();
			}
			return sb.toString();
		}

		public long nextLong() {
			if (!hasNext()) throw new NoSuchElementException();
			long n = 0;
			boolean minus = false;
			int b = readByte();
			if (b == '-') {
				minus = true;
				b = readByte();
			}
			if (b < '0' || '9' < b) {
				throw new NumberFormatException();
			}
			while (true) {
				if ('0' <= b && b <= '9') {
					n *= 10;
					n += b - '0';
				} else if (b == -1 || !isPrintableChar(b)) {
					return minus ? -n : n;
				} else {
					throw new NumberFormatException();
				}
				b = readByte();
			}
		}

		public int nextInt() {
			long nl = nextLong();
			if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE)
				throw new NumberFormatException();
			return (int) nl;
		}

		public double nextDouble() {
			return Double.parseDouble(next());
		}
	}
}