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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Linq;
using System.Text;
using System.Threading;
using static System.Console;
using static System.Math;

namespace AtTest.ForYellow
{
    class ABC160_F
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(OpenStandardOutput()) { AutoFlush = false };
            SetOut(sw);

            Method(args);

            Out.Flush();
        }

        static void Method(string[] args)
        {
            int n = ReadInt();
            List<int>[] graph = new List<int>[n];
            for(int i = 0; i < n; i++)
            {
                graph[i] = new List<int>();
            }

            for(int i = 0; i < n - 1; i++)
            {
                int[] ab = ReadInts();
                int a = ab[0] - 1;
                int b = ab[1] - 1;
                graph[a].Add(b);
                graph[b].Add(a);
            }

            int[] childs = new int[n];
            DFS1(0, -1, graph, ref childs);

            long mask = 1000000000 + 7;
            CaseCalculator calculator = new CaseCalculator(mask, n);

            long[] pats = new long[n];
            DFS2(0, -1, graph, calculator, ref pats, childs, mask);

            long[] res = new long[n];
            DFS3(0, -1, graph, res, childs, pats, calculator, mask);
            for(int i = 0; i < n; i++)
            {
                WriteLine(res[i]);
            }
        }

        static void DFS1(int now, int parent, List<int>[] graph, ref int[] childs)
        {
            int child = 0;
            for(int i = 0; i < graph[now].Count; i++)
            {
                int to = graph[now][i];
                if (to == parent)
                {
                    continue;
                }

                DFS1(to, now, graph, ref childs);
                child += childs[to] + 1;
            }

            childs[now] = child;
        }

        static void DFS2(int now,int parent, List<int>[] graph, CaseCalculator calculator,
            ref long[] pats,int[] childs,long mask)
        {
            long val = 1;
            int childCnt = childs[now];
            for (int i = 0; i < graph[now].Count; i++)
            {
                int to = graph[now][i];
                if (to == parent)
                {
                    continue;
                }

                long tmp = calculator.Combination(childCnt, childs[to] + 1);
                DFS2(to, now, graph, calculator, ref pats, childs, mask);
                tmp = (tmp * pats[to]) % mask;
                childCnt -= childs[to] + 1;
                val *= tmp;
                val %= mask;
            }

            pats[now] = val;
        }

        static void DFS3(int now, int parent,
            List<int>[] graph, long[] res,            int[] childs, 
            long[] pats, CaseCalculator calculator, long mask)
        {
            long pat = 1;
            if (parent == -1)
            {
                pat = pats[now];
            }
            else
            {
                long parentVal = (res[parent] * calculator.Inverse(
                    calculator.Permutation(childs.Length - 1, childs[now] + 1))) % mask;
                parentVal = (parentVal * calculator.Permutation(
                    childs[now] + 1, childs[now] + 1)) % mask;
                parentVal = (parentVal * calculator.Inverse(pats[now])) % mask;
                parentVal = (parentVal * calculator.Combination(
                    childs.Length - 1, childs[now])) % mask;
                pat = (parentVal * pats[now]) % mask;
            }
            res[now] = pat;

            for(int i = 0; i < graph[now].Count; i++)
            {
                int to = graph[now][i];
                if (to == parent)
                {
                    continue;
                }

                DFS3(to, now, graph, res, childs, pats, calculator, mask);
            }
             }

        class CaseCalculator
        {
            long mask;
            long[] allPermutations;
            long[] allInverses;

            public CaseCalculator(long mask, long permutationCnt)
            {
                this.mask = mask;
                allPermutations = AllPermutations(permutationCnt);
                allInverses = AllInverses(permutationCnt);
            }

            public long Combination(long n, long m)
            {
                if (n < m) return 0;

                if (n - m < m) m = n - m;

                return Multi(allPermutations[n], allInverses[n - m], allInverses[m]);
            }

            public long Permutation(long n, long m)
            {
                if (n < m) return 0;

                return Multi(allPermutations[n], allInverses[n - m]);
            }

            long[] AllPermutations(long n)
            {
                var perms = new long[n + 1];
                perms[0] = 1;
                for (int i = 1; i <= n; i++)
                {
                    perms[i] = Multi(perms[i - 1], i);
                }
                return perms;
            }

            long[] AllInverses(long n)
            {
                var inverses = new long[n + 1];
                inverses[1] = 1;
                var permInverses = new long[n + 1];
                permInverses[0] = 1;
                permInverses[1] = 1;
                for (int i = 2; i <= n; i++)
                {
                    inverses[i] = mask - inverses[mask % i] * (mask / i) % mask;
                    permInverses[i] = Multi(permInverses[i - 1], inverses[i]);
                }
                return permInverses;
            }

            public long Multi(params long[] vals)
            {
                if (vals.Length == 0)
                {
                    return 0;
                }

                long val = vals[0] % mask;
                for (int i = 1; i < vals.Length; i++)
                {
                    val *= (vals[i] % mask);
                    val %= mask;
                }
                return val;
            }

            public long Inverse(long val)
            {
                long a = mask;
                long b = val % mask;
                long x = 1;
                long x1 = 0;
                long y = 0;
                long y1 = 1;
                while (b > 0)
                {
                    long q = a / b;
                    long r = a % b;
                    long x2 = (mask + x - (q * x1) % mask) % mask;
                    long y2 = (mask + y - (q * y1) % mask) % mask;
                    a = b;
                    b = r;
                    x = x1;
                    x1 = x2;
                    y = y1;
                    y1 = y2;
                }
                return y;
            }

            long DeprecatedInverse(long val)
            {
                return Pow(val, mask - 2);
            }

            long Pow(long a, long exp)
            {
                if (exp == 0) return 1;
                else if (exp == 1) return a % mask;
                else
                {
                    long halfMod = Pow(a, exp / 2);
                    long nextMod = Multi(halfMod, halfMod);
                    if (exp % 2 == 0)
                    {
                        return nextMod;
                    }
                    else
                    {
                        return Multi(nextMod, a);
                    }
                }
            }
        }

        private static string Read() { return ReadLine(); }
        private static char[] ReadChars() { return Array.ConvertAll(Read().Split(), a => a[0]); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}