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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.util.List;
import java.io.BufferedWriter;
import java.io.Writer;
import java.io.OutputStreamWriter;
import java.util.InputMismatchException;
import java.io.IOException;
import java.io.InputStream;
import java.util.ArrayList;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 *
 * @author prakharjain
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        InputReader in = new InputReader(inputStream);
        OutputWriter out = new OutputWriter(outputStream);
        VSubtree solver = new VSubtree();
        solver.solve(1, in, out);
        out.close();
    }

    static class VSubtree {
        List[] g;
        int n;
        int m;
        long[] sub;
        long[] dp;
        long[] ans;

        public void solve(int testNumber, InputReader in, OutputWriter out) {
            n = in.nextInt();

            m = in.nextInt();

            g = new List[n];

            in.readUndirectedGraph(g, n, n - 1, 1);

            sub = new long[n];
            dp = new long[n];
            ans = new long[n];

            dfs(0, -1);

            dfs2(0, -1, -1, -1);

            for (int i = 0; i < n; i++) {
                out.println(ans[i]);
            }

        }

        void dfs(int u, int p) {
            sub[u] = 1;
            for (int i = 0; i < g[u].size(); i++) {
                int v = (int) g[u].get(i);

                if (v != p) {
                    dfs(v, u);
                    sub[u] *= (1 + sub[v]);
                    sub[u] %= m;
                }
            }
        }

        void dfs2(int u, int p, long prep, long sufp) {

            if (u == 0) {
                dp[0] = 1;
                ans[0] = sub[0];
            } else {
                long val = ((prep * sufp) % m) * dp[p];
                dp[u] = 1 + val;
                dp[u] %= m;

                ans[u] += (sub[u] * dp[u]) % m;
                ans[u] %= m;
            }

            int cc = g[u].size();

            if (u != 0) {
                cc--;
            }

            if (cc == 0)
                return;

            long[] val = new long[cc];
            long[] pre = new long[cc + 1];
            long[] suf = new long[cc + 2];

            for (int i = 0, j = 0; i < g[u].size(); i++) {
                int v = (int) g[u].get(i);

                if (v != p) {
                    val[j] = 1 + sub[v];
                    val[j] %= m;
                    j++;
                }
            }

            pre[0] = 1;

            for (int i = 0; i < cc; i++) {
                pre[i + 1] = pre[i] * val[i];
                pre[i + 1] %= m;
            }

            suf[cc + 1] = 1;

            for (int i = cc - 1; i >= 0; i--) {
                suf[i + 1] = suf[i + 2] * val[i];
                suf[i + 1] %= m;
            }

            for (int i = 0, j = 1; i < g[u].size(); i++) {
                int v = (int) g[u].get(i);

                if (v != p) {
                    dfs2(v, u, pre[j - 1], suf[j + 1]);
                    j++;
                }
            }
        }

    }

    static class InputReader {
        private InputStream stream;
        private byte[] buf = new byte[1024];
        private int curChar;
        private int numChars;
        private InputReader.SpaceCharFilter filter;

        public InputReader(InputStream stream) {
            this.stream = stream;
        }

        public static boolean isWhitespace(int c) {
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }

        public int read() {
            if (numChars == -1) {
                throw new InputMismatchException();
            }
            if (curChar >= numChars) {
                curChar = 0;
                try {
                    numChars = stream.read(buf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (numChars <= 0) {
                    return -1;
                }
            }
            return buf[curChar++];
        }

        public int nextInt() {
            int c = read();
            while (isSpaceChar(c)) {
                c = read();
            }
            int sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = read();
            }
            int res = 0;
            do {
                if (c < '0' || c > '9') {
                    throw new InputMismatchException();
                }
                res *= 10;
                res += c - '0';
                c = read();
            } while (!isSpaceChar(c));
            return res * sgn;
        }

        public boolean isSpaceChar(int c) {
            if (filter != null) {
                return filter.isSpaceChar(c);
            }
            return isWhitespace(c);
        }

        public void readUndirectedGraph(List[] g, int numOfVertices, int numOfEdges, int offset) {
            for (int i = 0; i < numOfVertices; i++) {
                g[i] = new ArrayList();
            }

            for (int i = 0; i < numOfEdges; i++) {
                int u = nextInt() - offset;
                int v = nextInt() - offset;

                g[u].add(v);
                g[v].add(u);
            }
        }

        public interface SpaceCharFilter {
            public boolean isSpaceChar(int ch);

        }

    }

    static class OutputWriter {
        private final PrintWriter writer;

        public OutputWriter(OutputStream outputStream) {
            writer = new PrintWriter(new BufferedWriter(new OutputStreamWriter(outputStream)));
        }

        public OutputWriter(Writer writer) {
            this.writer = new PrintWriter(writer);
        }

        public void close() {
            writer.close();
        }

        public void println(long i) {
            writer.println(i);
        }

    }
}