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.BufferedReader;
import java.io.InputStreamReader;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Scanner;

public class Main {
	public static void main(String[] args) {
		Scanner in = new Scanner(new BufferedReader(new InputStreamReader(System.in)));
		int n = in.nextInt();
		int m = in.nextInt();
		Map<Integer, List<Integer>> edges = getEdges(in, n - 1, false);
		long[] r = solve(n, m, edges);
		System.out.println(str(r));
		in.close();
	}

	private static long[] solve(int n, int m, Map<Integer, List<Integer>> edges) {
		long[] up = new long[n + 1];
		long[] sub = new long[n + 1];
		mod = m;
		dfs1(edges, sub, 0, 1);
		dfs3(edges, sub, up, new long[0], new long[0], 0, 1, 0);
		long[] r = new long[n];
		for (int i = 0; i < r.length; i++) {
			r[i] = mul(up[i + 1], sub[i + 1]);
		}
		return r;
	}

	private static long[] solve1(int n, int m, Map<Integer, List<Integer>> edges) {
		long[] dp = new long[n + 1];
		long[] sub = new long[n + 1];
		mod = m;
		dfs1(edges, sub, 0, 1);
		dfs2(edges, sub, dp, 0, 1);
		return Arrays.copyOfRange(dp, 1, dp.length);
	}

	static void dfs1(Map<Integer, List<Integer>> edges, long[] sub, int par, int cur) {
		sub[cur] = 1;
		List<Integer> adjV = edges.get(cur);
		if (adjV != null) {
			for (int v : adjV) {
				if (v != par) {
					dfs1(edges, sub, cur, v);
					sub[cur] = mul(sub[cur], (sub[v] + 1));
				}
			}
		}
	}

	static void dfs2(Map<Integer, List<Integer>> edges, long[] sub, long[] dp, int par, int cur) {
		dp[cur] = sub[cur] * (dp[par] / (sub[cur] + 1) + 1);
		List<Integer> adjV = edges.get(cur);
		if (adjV != null) {
			for (int v : adjV) {
				if (v != par) {
					dfs2(edges, sub, dp, cur, v);
				}
			}
		}
	}

	static void dfs3(Map<Integer, List<Integer>> edges, long[] sub, long[] up, long[] prefixProd, long[] suffixProd,
			int par, int cur, int i) {
		up[cur] = 1;
		if (i > 0) {
			up[cur] = mul(up[cur], prefixProd[i - 1]);
		}
		if (i < prefixProd.length - 1) {
			up[cur] = mul(up[cur], suffixProd[i + 1]);
		}
		up[cur] = add(mul(up[cur], up[par]), 1);
		List<Integer> adjV = edges.get(cur);
		if (adjV != null) {
			adjV.remove((Integer) par);
			prefixProd = new long[adjV.size()];
			suffixProd = new long[adjV.size()];
			for (int j = 0; j < adjV.size(); j++) {
				if (j == 0) {
					prefixProd[j] = add(sub[adjV.get(j)], 1);
					suffixProd[adjV.size() - j - 1] = add(sub[adjV.get(adjV.size() - j - 1)], 1);
				} else {
					prefixProd[j] = mul(add(sub[adjV.get(j)], 1), prefixProd[j - 1]);
					suffixProd[adjV.size() - j - 1] = mul(add(sub[adjV.get(adjV.size() - j - 1)], 1),
							suffixProd[adjV.size() - j]);
				}
			}
			for (int j = 0; j < adjV.size(); j++) {
				dfs3(edges, sub, up, prefixProd, suffixProd, cur, adjV.get(j), j);
			}
		}
	}

	static long mod = 1000000007;

	static long add(long a, long b) {
		long r = a + b;
		if (r < 0) {
			r += mod;
		}
		return r % mod;
	}

	static long mul(long a, long b) {
		return (a * b) % mod;
	}

	static String str(long[] a) {
		String[] str = new String[a.length];
		for (int i = 0; i < a.length; i++) {
			str[i] = a[i] + "";
		}
		return String.join(" ", str);
	}

	static int[] getIntArr(Scanner in, int size) {
		int[] arr = new int[size];
		for (int i = 0; i < size; i++) {
			arr[i] = in.nextInt();
		}
		return arr;
	}

	static int[][] getIntArr(Scanner in, int row, int col) {
		int[][] arr = new int[row][];
		for (int i = 0; i < row; i++) {
			arr[i] = getIntArr(in, col);
		}
		return arr;
	}

	static char[] getCharArr(Scanner in, int size) {
		char[] arr = in.next().toCharArray();
		return arr;
	}

	static String[] getStringArr(Scanner in, int size) {
		String[] arr = new String[size];
		for (int i = 0; i < size; i++) {
			arr[i] = in.next();
		}
		return arr;
	}

	static Map<Integer, List<Integer>> getEdges(Scanner in, int size, boolean directed) {
		Map<Integer, List<Integer>> edges = new HashMap<>();
		for (int i = 0; i < size; i++) {
			int from = in.nextInt();
			int to = in.nextInt();
			if (!edges.containsKey(from)) {
				edges.put(from, new ArrayList<Integer>());
			}
			edges.get(from).add(to);
			if (!directed) {
				if (!edges.containsKey(to)) {
					edges.put(to, new ArrayList<Integer>());
				}
				edges.get(to).add(from);
			}

		}
		return edges;
	}
}