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);

		// root を根としたときの1回目のDFS
		// iノードが黒の時のそこを根と見たときの組み合わせの数
		long[] dp = new long[n];
		for (int e : rt.postorder) {
			long tmp = 1;
			for (int next : rt.edges.get(e)) {
				if (next == rt.p[e]) {
					continue;
				}
				
				tmp *= dp[next] + 1;
				tmp %= m;

			}
			dp[e] = tmp;
		}

		// 各ノードでの答え
		long[] dp2 = Arrays.copyOf(dp, dp.length);
		// iノードの親ノードをiから見て子とみなしたときの値
		long[] fp = new long[n];
		for (int e : rt.preorder) {
			int p = rt.p[e];
			int size = rt.edges.get(e).size();
			
			// 累積積
			long[] cummul = new long[size + 1];
			cummul[0] = 1;
			for (int i = 0; i < size; i++) {
				int ee = rt.edges.get(e).get(i);
				// 親ノードの値はfp[e]を使う
				// 子ノードの値は1回目のDFSの値
				long tmp = dp[ee];
				if (ee == p) {
					tmp = fp[e];
				}
				cummul[i + 1] = cummul[i] * (tmp + 1) % m;
			}

			// 逆順累積積
			long[] cummul2 = new long[size + 1];
			cummul2[size] = 1;
			for (int i = size - 1; i >= 0; i--) {
				int ee = rt.edges.get(e).get(i);
				// 親ノードの値はfp[e]を使う
				// 子ノードの値は1回目のDFSの値
				long tmp = dp[ee];
				if (ee == p) {
					tmp = fp[e];
				}
				cummul2[i] = cummul2[i + 1] * (tmp + 1) % m;
			}
			
			for (int i = 0; i < size; i++) {
				int ee = rt.edges.get(e).get(i);
				// 子eeの値を除いて計算したものを子に伝搬
				long tmp = cummul[i] * cummul2[i + 1] % m;
				fp[ee] = tmp;
			}

			dp2[e] = cummul[size];
		}
		
		for (long e : dp2) {
			pr.println(e);
		}
	}

	/**
	 * 頂点と無向辺の情報から根付き木とみなして
	 * 親、深さ、行きがけ順、帰りがけ順を求める
	 */
	static class RoutedTree {
    	/** 頂点数 */
		int n;
    	/** 辺の個数 */
		int m;
    	/** 各頂点における辺情報 */
		List<List<Integer>> edges;

		/** 根 */
		int root;
		/** 各頂点における親 */
		int[] p;
		/** 各頂点の根から見た深さ */
		int[] d;
		/** 行きがけ順の頂点集合 */
		int[] preorder;
		/** 帰りがけ順の頂点集合 */
		int[] postorder;
		
		/**
		 * n 個の頂点を管理するインスタンスを返す
		 * 
		 * @param n 頂点の数
		 */
		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];
			postorder = new int[n];
		}

		/**
		 * uからvへの辺を追加する
		 * n-1個の辺が追加され、連結なグラフであることを期待する
		 * 
		 * @param u {@literal 0<=u<n}
		 * @param v {@literal 0<=v<n}
		 */
		void addEdge(int u, int v) {
			edges.get(u).add(v);
			edges.get(v).add(u);
			m++;
		}

		/**
		 * root を根とした根付き木をDFSする
		 * 
		 * @param root 根と見なす頂点
		 */
		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;
			int posti = 0;
			while (!st.isEmpty()) {
				int e = st.peek();
				
				if (used[e]) {
					// postorder
					
					postorder[posti++] = e;
					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);
		}
	}
}