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 {

	private static Scanner sc;
	private static Printer pr;

	private static void solve() {
		int n = sc.nextInt();
		int m = sc.nextInt();

		int[][] xy = sc.nextIntArrays(2, n - 1);

		RoutedTree rt = new RoutedTree(n);
		for (int i = 0; i < n - 1; i++) {
			int x = xy[0][i] - 1;
			int y = xy[1][i] - 1;

			rt.addEdge(x, y);
		}

		int root = 0;
		rt.dfs(root);

		List<List<Long>> dp3 = new ArrayList<>(n);
		List<List<Long>> dp4 = new ArrayList<>(n);
		List<List<Long>> dp5 = new ArrayList<>(n);
		for (int i = 0; i < n; i++) {
			dp3.add(new ArrayList<>());
		}
		for (int i = 0; i < n; i++) {
			dp4.add(new ArrayList<>());
			dp5.add(new ArrayList<>());
		}
		
		long[] dp = new long[n];
		for (int i = n - 1; i >= 0; i--) {
			int e = rt.preorder[i];
			
			long tmp = 1;
			for (int next : rt.edges.get(e)) {
				if (next == rt.p[e]) {
					continue;
				}
				
				tmp *= dp[next] + 1;
				tmp %= m;

				if (i == root) {
					dp3.get(i).add(dp[next]);
				}
			}
			dp[e] = tmp;
		}
		
		long[] dp2 = Arrays.copyOf(dp, dp.length);
		for (int e : rt.preorder) {
			if (e == root) {
				dp4.get(e).add(1L);
				for (int i = 0, size = rt.edges.get(e).size(); i < size; i++) {
					dp4.get(e).add(dp4.get(e).get(i) * (dp3.get(e).get(i) + 1) % m);
				}
				for (int i = 0, size = rt.edges.get(e).size(); i < size + 1; i++) {
					dp5.get(e).add(1L);
				}
				for (int i = rt.edges.get(e).size() - 1; i >= 0; i--) {
					dp5.get(e).set(i, dp5.get(e).get(i + 1) * (dp3.get(e).get(i) + 1) % m);
				}
				
				continue;
			}
			
			int p = rt.p[e];

			long tmp = 1;
			/*
			for (int i = 0, size = rt.edges.get(p).size(); i < size; i++) {
				int c = rt.edges.get(p).get(i);

				if (c == e) {
					continue;
				}
				
				tmp *= dp3.get(p).get(i) + 1;
				tmp %= m;
			}
			*/
			int index = Collections.binarySearch(rt.edges.get(p), e);
			tmp *= dp4.get(p).get(index);
			tmp %= m;
			tmp *= dp5.get(p).get(index + 1);
			tmp %= m;

			for (int i = 0, size = rt.edges.get(e).size(); i < size; i++) {
				int c = rt.edges.get(e).get(i);
				
				if (c == p) {
					dp3.get(e).add(tmp);
				} else {
					dp3.get(e).add(dp[c]);
				}
			}
			dp4.get(e).add(1L);
			for (int i = 0, size = rt.edges.get(e).size(); i < size; i++) {
				dp4.get(e).add(dp4.get(e).get(i) * (dp3.get(e).get(i) + 1) % m);
			}
			for (int i = 0, size = rt.edges.get(e).size(); i < size + 1; i++) {
				dp5.get(e).add(1L);
			}
			for (int i = rt.edges.get(e).size() - 1; i >= 0; i--) {
				dp5.get(e).set(i, dp5.get(e).get(i + 1) * (dp3.get(e).get(i) + 1) % m);
			}

			dp2[e] *= tmp + 1;
			dp2[e] %= m;
		}
		
		for (long e : dp2) {
			pr.println(e);
		}
	}

	static class RoutedTree {
		int n;
		int m;
		List<List<Integer>> edges;
		
		int root;
		int[] p;
		int[] d;
		int[] preorder;
		
		RoutedTree(int n) {
			this.n = n;
			edges = new ArrayList<>(n);
			for (int i = 0; i < n; i++) {
				edges.add(new ArrayList<>());
			}

			root = -1;
			p = new int[n];
			d = new int[n];
			preorder = new int[n];
		}

		void addEdge(int u, int v) {
			edges.get(u).add(v);
			edges.get(v).add(u);
			m++;
		}

		void dfs(int root) {
			if (this.root == root) {
				return;
			}

			for (int i = 0; i < n; i++) {
				Collections.sort(edges.get(i));
			}
			
			this.root = root;
			Arrays.fill(p, 0);
			Arrays.fill(d, 0);
			boolean[] used = new boolean[n];
			Deque<Integer> st = new ArrayDeque<>();
			st.push(root);
			p[root] = -1;
			d[root] = 0;
			int prei = 0;
			while (!st.isEmpty()) {
				int e = st.peek();
				
				if (used[e]) {
					// postorder
					
					st.pop();
				} else {
					// preorder
					for (int next : edges.get(e)) {
						if (next == p[e]) {
							continue;
						}

						p[next] = e;
						d[next] = d[e] + 1;
						st.push(next);
					}
					
					used[e] = true;
					preorder[prei++] = e;
				}
			}
		}
	}
	
	// ---------------------------------------------------
	public static void main(String[] args) {
		sc = new Scanner(System.in);
		pr = new Printer(System.out);
			
		solve();
			
		pr.close();
		sc.close();
	}

	static class Scanner {
		BufferedReader br;

		Scanner (InputStream in) {
			br = new BufferedReader(new InputStreamReader(in));
		}

		private boolean isPrintable(int ch) {
			return ch >= '!' && ch <= '~';
		}

		private boolean isCRLF(int ch) {
			return ch == '\n' || ch == '\r' || ch == -1;
		}

		private int nextPrintable() {
			try {
				int ch;
				while (!isPrintable(ch = br.read())) {
					if (ch == -1) {
						throw new NoSuchElementException();
					}
				}

				return ch;
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}

		String next() {
			try {
				int ch = nextPrintable();
				StringBuilder sb = new StringBuilder();
				do {
					sb.appendCodePoint(ch);
				} while (isPrintable(ch = br.read()));

				return sb.toString();
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}

		int nextInt() {
			try {
				// parseInt from Integer.parseInt()
				boolean negative = false;
				int res = 0;
				int limit = -Integer.MAX_VALUE;
				int radix = 10;

				int fc = nextPrintable();
				if (fc < '0') {
					if (fc == '-') {
						negative = true;
						limit = Integer.MIN_VALUE;
					} else if (fc != '+') {
						throw new NumberFormatException();
					}
					fc = br.read();
				}
				int multmin = limit / radix;

				int ch = fc;
				do {
					int digit = ch - '0';
					if (digit < 0 || digit >= radix) {
						throw new NumberFormatException();
					}
					if (res < multmin) {
						throw new NumberFormatException();
					}
					res *= radix;
					if (res < limit + digit) {
						throw new NumberFormatException();
					}
					res -= digit;

				} while (isPrintable(ch = br.read()));

				return negative ? res : -res;
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}

		long nextLong() {
			try {
				// parseLong from Long.parseLong()
				boolean negative = false;
				long res = 0;
				long limit = -Long.MAX_VALUE;
				int radix = 10;

				int fc = nextPrintable();
				if (fc < '0') {
					if (fc == '-') {
						negative = true;
						limit = Long.MIN_VALUE;
					} else if (fc != '+') {
						throw new NumberFormatException();
					}
					fc = br.read();
				}
				long multmin = limit / radix;

				int ch = fc;
				do {
					int digit = ch - '0';
					if (digit < 0 || digit >= radix) {
						throw new NumberFormatException();
					}
					if (res < multmin) {
						throw new NumberFormatException();
					}
					res *= radix;
					if (res < limit + digit) {
						throw new NumberFormatException();
					}
					res -= digit;

				} while (isPrintable(ch = br.read()));

				return negative ? res : -res;
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}

		float nextFloat() {
			return Float.parseFloat(next());
		}

		double nextDouble() {
			return Double.parseDouble(next());
		}

		String nextLine() {
			try {
				int ch;
				while (isCRLF(ch = br.read())) {
					if (ch == -1) {
						throw new NoSuchElementException();
					}
				}
				StringBuilder sb = new StringBuilder();
				do {
					sb.appendCodePoint(ch);
				} while (!isCRLF(ch = br.read()));

				return sb.toString();
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}
		
		int[] nextIntArray(int n) {
			int[] ret = new int[n];
			for (int i = 0; i < n; i++) {
				ret[i] = sc.nextInt();
			}
			
			return ret;
		}

		long[] nextLongArray(int n) {
			long[] ret = new long[n];
			for (int i = 0; i < n; i++) {
				ret[i] = sc.nextLong();
			}
			
			return ret;
		}

		int[][] nextIntArrays(int m, int n) {
			int[][] ret = new int[m][n];
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < m; j++) {
					ret[j][i] = sc.nextInt();
				}
			}
			
			return ret;
		}

		void close() {
			try {
				br.close();
			} catch (IOException e) {
//				throw new NoSuchElementException();
			}
		}
	}

	static class Printer extends PrintWriter {
		Printer(OutputStream out) {
			super(out);
		}
		
		void printInts(int... a) {
			StringBuilder sb = new StringBuilder(32);
			for (int i = 0, size = a.length; i < size; i++) {
				if (i > 0) {
					sb.append(' ');
				}
				sb.append(a[i]);
			}

			println(sb);
		}
		
		void printLongs(long... a) {
			StringBuilder sb = new StringBuilder(64);
			for (int i = 0, size = a.length; i < size; i++) {
				if (i > 0) {
					sb.append(' ');
				}
				sb.append(a[i]);
			}

			println(sb);
		}
		
		void printStrings(String... a) {
			StringBuilder sb = new StringBuilder(32);
			for (int i = 0, size = a.length; i < size; i++) {
				if (i > 0) {
					sb.append(' ');
				}
				sb.append(a[i]);
			}

			println(sb);
		}
	}
}