Problem:
Score : 100 points

Problem Statement

There is a tree with N vertices, numbered 1, 2, \ldots, N.
For each i (1 \leq i \leq N - 1), the i-th edge connects Vertex x_i and y_i.

Taro has decided to paint each vertex in white or black, so that any black vertex can be reached from any other black vertex by passing through only black vertices.

You are given a positive integer M.
For each v (1 \leq v \leq N), answer the following question:

Assuming that Vertex v has to be black, find the number of ways in which the vertices can be painted, modulo M.

Constraints

All values in input are integers.

1 \leq N \leq 10^5

2 \leq M \leq 10^9

1 \leq x_i, y_i \leq N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N M
x_1 y_1
x_2 y_2
:
x_{N - 1} y_{N - 1}

Output

Print N lines.
The v-th (1 \leq v \leq N) line should contain the answer to the following question:

Assuming that Vertex v has to be black, find the number of ways in which the vertices can be painted, modulo M.

Sample Input 1

3 100
1 2
2 3

Sample Output 1

3
4
3

There are seven ways to paint the vertices, as shown in the figure below.
Among them, there are three ways such that Vertex 1 is black, four ways such that Vertex 2 is black and three ways such that Vertex 3 is black.

Sample Input 2

4 100
1 2
1 3
1 4

Sample Output 2

8
5
5
5

Sample Input 3

1 100

Sample Output 3

1

Sample Input 4

10 2
8 5
10 8
6 5
1 5
4 8
2 10
3 6
9 2
1 7

Sample Output 4

0
0
1
1
1
0
1
0
1
1

Be sure to print the answers modulo M.

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Text;
using System.Globalization;
using System.Diagnostics;



class Myon
{
    public Myon() { }
    public static int Main()
    {
        new Myon().calc();
        return 0;
    }

    Scanner cin;

    int N;

    long[,] dp;
    List<int>[] es;
    int[] A;
    int[] B;

    long mod;

    int[] NCNT;
    long[][] muls;

    int[,] pos;

    void calc()
    {
        cin = new Scanner();
        N = cin.nextInt();
        mod = cin.nextInt();

        es = new List<int>[N];
        for (int i = 0; i < N; i++)
        {
            es[i] = new List<int>();
        }
        A = new int[N - 1];
        B = new int[N - 1];
        pos = new int[N - 1, 2];

        for (int i = 0; i < N - 1; i++)
        {
            A[i] = cin.nextInt() - 1;
            B[i] = cin.nextInt() - 1;

            pos[i, 0] = es[B[i]].Count;
            pos[i, 1] = es[A[i]].Count;

            es[A[i]].Add(i);
            es[B[i]].Add(i);
        }

        dp = new long[N - 1, 2];
        for (int i = 0; i < N - 1; i++)
        {
            for (int j = 0; j < 2; j++)
            {
                dp[i, j] = -1;
            }
        }

        NCNT = new int[N];
        muls = new long[N][];

        for (int i = 0; i < N; i++)
        {
            long p = 1;
            foreach (var e2 in es[i])
            {
                if (A[e2] == i) p *= (dfs(e2, 0) + 1);
                else p *= dfs(e2, 1) + 1;
                p %= mod;

            }
            Console.WriteLine(p);
        }

    }


    long dfs(int e, int t)
    {
        if (dp[e, t] != -1) return dp[e, t];
        int pre, a;
        if (t == 0)
        {
            pre = A[e];
            a = B[e];
        }
        else
        {
            pre = B[e];
            a = A[e];
        }



        NCNT[a]++;
        if (NCNT[a] >= 3 && es[a].Count >= 10)
        {
            int len = (int)Math.Sqrt(es[a].Count);
            int N = (es[a].Count + len - 1) / len;
            if (NCNT[a] == 3)
            {
                muls[a] = new long[N];
                for (int i = 0; i < N; i++)
                {
                    muls[a][i] = 1;
                }

                for (int i = 0; i < es[a].Count; i++)
                {
                    int e2 = es[a][i];
                    long pp;
                    if (A[e2] == a) pp = (dfs(e2, 0) + 1);
                    else pp = dfs(e2, 1) + 1;
                    muls[a][i / len] *= pp;
                    muls[a][i / len] %= mod;
                }
            }


            long p = 1;
            for (int i = 0; i < N; i++)
            {
                if (pos[e, t] / len != i)
                {
                    p *= muls[a][i];
                    p %= mod;
                }
                else
                {
                    for (int j = i * len; j < Math.Min((i + 1) * len, es[a].Count); j++)
                    {
                        int e2 = es[a][j];
                        if (e2 == e) continue;
                        if (A[e2] == a) p *= (dfs(e2, 0) + 1);
                        else p *= dfs(e2, 1) + 1;
                        p %= mod;
                    }
                }
            }

            return dp[e, t] = p;
        }
        else
        {
            long p = 1;
            foreach (var e2 in es[a])
            {
                if (e2 == e) continue;
                if (A[e2] == a) p *= (dfs(e2, 0) + 1);
                else p *= dfs(e2, 1) + 1;
                p %= mod;

            }
            return dp[e, t] = p;
        }
    }



}


class Scanner
{
    string[] s;
    int i;

    char[] cs = new char[] { ' ' };

    public Scanner()
    {
        s = new string[0];
        i = 0;
    }

    public string next()
    {
        if (i < s.Length) return s[i++];
        string st = Console.ReadLine();
        while (st == "") st = Console.ReadLine();
        s = st.Split(cs, StringSplitOptions.RemoveEmptyEntries);
        if (s.Length == 0) return next();
        i = 0;
        return s[i++];
    }

    public int nextInt()
    {
        return int.Parse(next());
    }
    public int[] ArrayInt(int N, int add = 0)
    {
        int[] Array = new int[N];
        for (int i = 0; i < N; i++)
        {
            Array[i] = nextInt() + add;
        }
        return Array;
    }

    public long nextLong()
    {
        return long.Parse(next());
    }

    public long[] ArrayLong(int N, long add = 0)
    {
        long[] Array = new long[N];
        for (int i = 0; i < N; i++)
        {
            Array[i] = nextLong() + add;
        }
        return Array;
    }

    public double nextDouble()
    {
        return double.Parse(next());
    }


    public double[] ArrayDouble(int N, double add = 0)
    {
        double[] Array = new double[N];
        for (int i = 0; i < N; i++)
        {
            Array[i] = nextDouble() + add;
        }
        return Array;
    }
}

class XRand
{
    uint x, y, z, w;


    public XRand()
    {
        init();
    }

    public XRand(uint s)
    {
        init();
        init_xor128(s);
    }

    void init()
    {
        x = 314159265; y = 358979323; z = 846264338; w = 327950288;

    }

    public void init_xor128(uint s)
    {
        z ^= s;
        z ^= z >> 21; z ^= z << 35; z ^= z >> 4;
        z *= 736338717;
    }

    uint next()
    {
        uint t = x ^ x << 11; x = y; y = z; z = w; return w = w ^ w >> 19 ^ t ^ t >> 8;
    }

    public long nextLong(long m)
    {
        return (long)((((ulong)next() << 32) + next()) % (ulong)m);
    }

    public int nextInt(int m)
    {
        return (int)(next() % m);
    }

    public int nextIntP(int a)
    {
        return (int)Math.Pow(a, nextDouble());
    }

    public int nextInt(int min, int max)
    {
        return min + nextInt(max - min + 1);
    }


    public double nextDouble()
    {
        return (double)next() / uint.MaxValue;
    }

    public double nextDoubleP(double a)
    {
        return Math.Pow(a, nextDouble());
    }
}