Problem: 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.

Examples

Input

3 100
1 2
2 3


Output

3
4
3


Input

4 100
1 2
1 3
1 4


Output

8
5
5
5


Input

1 100


Output

1


Input

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


Output

0
0
1
1
1
0
1
0
1
1

Solution (in JAVA):

Solution:
import java.io.*;
import java.util.*;

public class Main {
    public static void main(String[] args) throws IOException {
        Scan sc = new Scan();
        int n = sc.scanInt();
        int m = sc.scanInt();
        Sol sol = new Sol(n, m);
        while(n-- > 1){
            sol.ae(sc.scanInt(),sc.scanInt());
        }
        sol.sol();
    }
}

class Sol {
    int n, mod;
    ArrayList<Integer>[] adj;
    int[] in, out, ans, sib;
    
    Sol(int n, int m){
        this.n = n;
        this.mod = m;
        adj = new ArrayList[n+1];
        in = new int[n+1];
        out = new int[n+1];
        ans = new int[n+1];
        sib = new int[n+1];
        for(int i = 0; i <= n; ++i)
            adj[i] = new ArrayList<Integer>();
    }
    
    void ae(int a, int b){
        adj[a].add(b);
        adj[b].add(a);
    }
    
    void dfs1(int src, int par){
        // in[src] = #ways to paint this node black while keeping black nodes connected within it's subtree
        long res = 1;
        if(par == 0)
            sib[src] = 1;
        int sibs = adj[src].size();
        int sz = sibs;
        if(par != 0)
            sz--;
        int[] left_sibs = new int[sz];
        int[] right_sibs = new int[sz];
        int i = 0;
        for(int nbr : adj[src]){
            if(nbr == par)
                continue;
            dfs1(nbr, src);
            // 1 is added since there is one way to paint the subtree rooted at nbr as all white
            // since all black nodes need to be connected, if nbr is white, then the subtree rooted at nbr is all white
            res *= ((long)in[nbr]+1);
            res %= mod;
            left_sibs[i++] = (int) res;
        }
        in[src] = (int) res;
        res = 1;
        for(int it = sibs-1; it >= 0; --it){
            int nd = adj[src].get(it);
            if(nd == par)
                continue;
            res *= (in[nd]+1);
            res %= mod;
            right_sibs[--i] = (int) res;
        }
        i = 0;
        for(int it = 0; it < sibs; ++it){
            int nd = adj[src].get(it);
            if(nd == par)
                continue;
            res = 1;
            if(i > 0)
                res *= left_sibs[i-1];
            res %= mod;
            if(i < sz-1)
                res *= right_sibs[i+1];
            res %= mod;
            sib[nd] = (int) res;
            ++i;
        }
    }
    
    void dfs2(int src, int par){
        long res = 1;
        if(par == 0)
            out[src] = 0;
        else {
            res = ((long) out[par] + 1)* sib[src];
            out[src] = (int)(res%mod);
        }
        res = in[src] * ((long) out[src] + 1);
        ans[src] = (int)(res%mod);
        for(int nbr : adj[src]){
            if(nbr == par)
                continue;
            dfs2(nbr, src);
        }        
    }
    
    void sol() {
        dfs1(1, 0);
        dfs2(1, 0);
        for(int i = 1; i <= n; ++i)
            System.out.println(ans[i]);
    }
}

class Scan {
    private byte[] buf=new byte[1024];
    private int index;
    private InputStream in;
    private int total;
    public Scan()
    {
        in=System.in;
    }
    public int scan()throws IOException
    {
        if(total<0)
        throw new InputMismatchException();
        if(index>=total)
        {
            index=0;
            total=in.read(buf);
            if(total<=0)
            return -1;
        }
        return buf[index++];
    }
    public int scanInt()throws IOException
    {
        int integer=0;
        int n=scan();
        while(isWhiteSpace(n))
        n=scan();
        int neg=1;
        if(n=='-')
        {
            neg=-1;
            n=scan();
        }
        while(!isWhiteSpace(n))
        {
            if(n>='0'&&n<='9')
            {
                integer*=10;
                integer+=n-'0';
                n=scan();
            }
            else throw new InputMismatchException();
        }
        return neg*integer;
    }
    public double scanDouble()throws IOException
    {
        double doub=0;
        int n=scan();
        while(isWhiteSpace(n))
        n=scan();
        int neg=1;
        if(n=='-')
        {
            neg=-1;
            n=scan();
        }
        while(!isWhiteSpace(n)&&n!='.')
        {
            if(n>='0'&&n<='9')
            {
                doub*=10;
                doub+=n-'0';
                n=scan();
            }
            else throw new InputMismatchException();
        }
        if(n=='.')
        {
            n=scan();
            double temp=1;
            while(!isWhiteSpace(n))
            {
                if(n>='0'&&n<='9')
                {
                    temp/=10;
                    doub+=(n-'0')*temp;
                    n=scan();
                }
                else throw new InputMismatchException();
            }
        }
        return doub*neg;
    }
    public String scanString()throws IOException
    {
        StringBuilder sb=new StringBuilder();
        int n=scan();
        while(isWhiteSpace(n))
        n=scan();
        while(!isWhiteSpace(n))
        {
            sb.append((char)n);
            n=scan();
        }
        return sb.toString();
    }
    private boolean isWhiteSpace(int n)
    {
        if(n==' '||n=='\n'||n=='\r'||n=='\t'||n==-1)
        return true;
        return false;
    }
}