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.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.IOException;
import java.util.ArrayList;
import java.io.UncheckedIOException;
import java.util.List;
import java.io.Closeable;
import java.io.Writer;
import java.io.OutputStreamWriter;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 */
public class Main {
    public static void main(String[] args) throws Exception {
        Thread thread = new Thread(null, new TaskAdapter(), "", 1 << 27);
        thread.start();
        thread.join();
    }

    static class TaskAdapter implements Runnable {
        @Override
        public void run() {
            InputStream inputStream = System.in;
            OutputStream outputStream = System.out;
            FastInput in = new FastInput(inputStream);
            FastOutput out = new FastOutput(outputStream);
            TaskV solver = new TaskV();
            solver.solve(1, in, out);
            out.close();
        }
    }

    static class TaskV {
        Modular mod;

        public void solve(int testNumber, FastInput in, FastOutput out) {
            int n = in.readInt();
            int m = in.readInt();
            mod = new Modular(m);
            Node[] nodes = new Node[n + 1];
            for (int i = 1; i <= n; i++) {
                nodes[i] = new Node();
            }
            for (int i = 1; i < n; i++) {
                Node a = nodes[in.readInt()];
                Node b = nodes[in.readInt()];
                a.next.add(b);
                b.next.add(a);
            }
            dfs(nodes[1], null);
            answer(nodes[1], null);
            for (int i = 1; i <= n; i++) {
                out.println(nodes[i].ans);
            }
        }

        public void dfs(Node root, Node p) {
            root.next.remove(p);
            for (Node node : root.next) {
                dfs(node, root);
            }
            root.dp[0] = 1;
            root.dp[1] = 1;
            for (Node node : root.next) {
                root.dp[1] = mod.mul(root.dp[1], mod.plus(node.dp[0], node.dp[1]));
            }
            for (Node node : root.next) {
                root.dp[2] = mod.plus(root.dp[2], mod.plus(node.dp[1], node.dp[2]));
            }
        }

        public void answer(Node root, Node p) {
            int dp1 = 1;
            int dp2 = 0;
            if (p != null) {
                dp1 = mod.plus(p.dp[0], p.dp[1]);
                dp2 = mod.plus(p.dp[1], p.dp[2]);
            }
            root.ans = mod.mul(root.dp[1], dp1);
            int originDp2 = mod.plus(root.dp[2], dp2);

            int[] pre = new int[root.next.size()];
            int[] post = new int[root.next.size()];
            for (int i = 0; i < root.next.size(); i++) {
                Node node = root.next.get(i);
                pre[i] = mod.plus(node.dp[0], node.dp[1]);
                if (i > 0) {
                    pre[i] = mod.mul(pre[i], pre[i - 1]);
                }
            }
            for (int i = root.next.size() - 1; i >= 0; i--) {
                Node node = root.next.get(i);
                post[i] = mod.plus(node.dp[0], node.dp[1]);
                if (i + 1 < root.next.size()) {
                    post[i] = mod.mul(post[i], post[i + 1]);
                }
            }
            for (int i = 0; i < root.next.size(); i++) {
                Node node = root.next.get(i);
                root.dp[1] = dp1;
                if (i > 0) {
                    root.dp[1] = mod.mul(root.dp[1], pre[i - 1]);
                }
                if (i + 1 < root.next.size()) {
                    root.dp[1] = mod.mul(root.dp[1], post[i + 1]);
                }
                root.dp[2] = mod.subtract(originDp2, mod.plus(node.dp[1], node.dp[2]));
                answer(node, root);
            }
        }

    }

    static class Node {
        List<Node> next = new ArrayList<>();
        int[] dp = new int[3];
        int ans;

    }

    static class FastOutput implements AutoCloseable, Closeable {
        private StringBuilder cache = new StringBuilder(10 << 20);
        private final Writer os;

        public FastOutput(Writer os) {
            this.os = os;
        }

        public FastOutput(OutputStream os) {
            this(new OutputStreamWriter(os));
        }

        public FastOutput println(int c) {
            cache.append(c).append('\n');
            return this;
        }

        public FastOutput flush() {
            try {
                os.append(cache);
                os.flush();
                cache.setLength(0);
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
            return this;
        }

        public void close() {
            flush();
            try {
                os.close();
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
        }

        public String toString() {
            return cache.toString();
        }

    }

    static class Modular {
        int m;

        public Modular(int m) {
            this.m = m;
        }

        public Modular(long m) {
            this.m = (int) m;
            if (this.m != m) {
                throw new IllegalArgumentException();
            }
        }

        public Modular(double m) {
            this.m = (int) m;
            if (this.m != m) {
                throw new IllegalArgumentException();
            }
        }

        public int valueOf(int x) {
            x %= m;
            if (x < 0) {
                x += m;
            }
            return x;
        }

        public int valueOf(long x) {
            x %= m;
            if (x < 0) {
                x += m;
            }
            return (int) x;
        }

        public int mul(int x, int y) {
            return valueOf((long) x * y);
        }

        public int plus(int x, int y) {
            return valueOf(x + y);
        }

        public int subtract(int x, int y) {
            return valueOf(x - y);
        }

        public String toString() {
            return "mod " + m;
        }

    }

    static class FastInput {
        private final InputStream is;
        private byte[] buf = new byte[1 << 13];
        private int bufLen;
        private int bufOffset;
        private int next;

        public FastInput(InputStream is) {
            this.is = is;
        }

        private int read() {
            while (bufLen == bufOffset) {
                bufOffset = 0;
                try {
                    bufLen = is.read(buf);
                } catch (IOException e) {
                    bufLen = -1;
                }
                if (bufLen == -1) {
                    return -1;
                }
            }
            return buf[bufOffset++];
        }

        public void skipBlank() {
            while (next >= 0 && next <= 32) {
                next = read();
            }
        }

        public int readInt() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            int val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

    }
}