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.*;

class Solver {
  final int n;
  final long m;
  final int[][] xys;
  
  List<List<Integer>> graph;
  List<Map<Integer, Integer>> graphChildIndexMap;
  
  ModCalculator mc;
  long[][] countMatrix;
  long[][] countMulFromHeadMatrix;
  long[][] countMulFromTailMatrix;
  
  Solver(int n, long m, int[][] xys) {
    this.n = n;
    this.m = m;
    this.xys = xys;
  }
  
  private long calculateCountMulFromHead(int index, int childIndexPosition) {
    if (childIndexPosition < 0) {
      return 1L;
    }
    if (countMulFromHeadMatrix[index][childIndexPosition] >= 0) {
      return countMulFromHeadMatrix[index][childIndexPosition];
    }
    
    int tempChildIndexPosition = childIndexPosition;
    while (tempChildIndexPosition >= 0 && countMulFromHeadMatrix[index][tempChildIndexPosition] < 0) {
      tempChildIndexPosition--;
    }
    for (int i = tempChildIndexPosition + 1; i <= childIndexPosition; i++) {
      long cacheValue = countMatrix[index][i];
      if (cacheValue < 0) {
        int childIndex = graph.get(index).get(i);
        cacheValue = mc.add(calculateCount(childIndex, index), 1);
        countMatrix[index][i] = cacheValue;
      }
      long v = i - 1 < 0 ? 1 : countMulFromHeadMatrix[index][i - 1];
      long countMulValue = mc.mul(cacheValue, v);
      countMulFromHeadMatrix[index][i] = countMulValue;
    }
    return countMulFromHeadMatrix[index][childIndexPosition];
  }
  
  private long calculateCountMulFromTail(int index, int childIndexPosition) {
    if (childIndexPosition >= graph.get(index).size()) {
      return 1L;
    }
    long cacheMulValue = countMulFromTailMatrix[index][childIndexPosition];
    if (cacheMulValue >= 0) {
      return cacheMulValue;
    }
    int tempChildIndexPosition = childIndexPosition;
    while (tempChildIndexPosition < graph.get(index).size() && countMulFromTailMatrix[index][tempChildIndexPosition] < 0) {
      tempChildIndexPosition++;
    }
    for (int i = tempChildIndexPosition - 1; i >= childIndexPosition; i--) {
      long cacheValue = countMatrix[index][i];
      if (cacheValue < 0) {
        int childIndex = graph.get(index).get(i);
        cacheValue = mc.add(calculateCount(childIndex, index), 1);
        countMatrix[index][i] = cacheValue;
      }
      long v = i + 1 >= graph.get(index).size() ? 1 : countMulFromTailMatrix[index][i + 1];
      long countMulValue = mc.mul(cacheValue, v);
      countMulFromTailMatrix[index][i] = countMulValue;
    }
    return countMulFromTailMatrix[index][childIndexPosition];
  }
  
  private long calculateCount(int index, int parentIndex) {
    int parentIndexPosition = graphChildIndexMap.get(index).get(parentIndex);
    long v1 = calculateCountMulFromHead(index, parentIndexPosition - 1);
    long v2 = calculateCountMulFromTail(index, parentIndexPosition + 1);
    return mc.mul(v1, v2);
  }
  
  private long solve(int index) {
    return calculateCountMulFromHead(index, graph.get(index).size() - 1);
  }
  
  public List<Long> solve() {
    graph = new ArrayList<>();
    graphChildIndexMap = new ArrayList<>();
    for (int i = 0; i <= n; i++) {
      graph.add(new ArrayList<>());
      graphChildIndexMap.add(new HashMap<>());
    }
    for (int[] xy : xys) {
      int x = xy[0];
      int y = xy[1];
      graphChildIndexMap.get(x).put(y, graph.get(x).size());
      graph.get(x).add(y);
      graphChildIndexMap.get(y).put(x, graph.get(y).size());
      graph.get(y).add(x);
    }
    
    countMatrix = new long[n + 1][];
    countMulFromHeadMatrix = new long[n + 1][];
    countMulFromTailMatrix = new long[n + 1][];
    for (int i = 0; i <= n; i++) {
      countMatrix[i] = new long[graph.get(i).size()];
      countMulFromHeadMatrix[i] = new long[graph.get(i).size()];
      countMulFromTailMatrix[i] = new long[graph.get(i).size()];
      Arrays.fill(countMatrix[i], -1);
      Arrays.fill(countMulFromHeadMatrix[i], -1);
      Arrays.fill(countMulFromTailMatrix[i], -1);
    }
    
    mc = new ModCalculator(m);
    
    List<Long> answers = new ArrayList<>();
    for (int i = 1; i <= n; i++) {
      answers.add(solve(i));
    }
    return answers;
  }
}

public class Main {
  private static void execute(ContestReader reader, PrintWriter out) {
    int n = reader.nextInt();
    long m = reader.nextLong();
    int[][] xys = reader.nextInt(n - 1, 2);
    for (long answer : new Solver(n, m, xys).solve()) {
      out.println(answer);
    }
  }
  
  public static void main(String[] args) {
    ContestReader reader = new ContestReader(System.in);
    PrintWriter out = new PrintWriter(System.out);
    execute(reader, out);
    out.flush();
  }
}

class ContestReader {
  private BufferedReader reader;
  private StringTokenizer tokenizer;
  
  ContestReader(InputStream in) {
    reader = new BufferedReader(new InputStreamReader(in));
  }
  
  public String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }
  
  public char nextChar() {
    return next().charAt(0);
  }
  
  public int nextInt() {
    return Integer.parseInt(next());
  }
  
  public long nextLong() {
    return Long.parseLong(next());
  }
  
  public double nextDouble() {
    return Double.parseDouble(next());
  }
  
  public String[] next(int n) {
    String[] array = new String[n];
    for (int i = 0; i < n; i++) {
      array[i] = next();
    }
    return array;
  }
  
  public char[] nextChar(int n) {
    char[] array = new char[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextChar();
    }
    return array;
  }
  
  public int[] nextInt(int n) {
    int[] array = new int[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextInt();
    }
    return array;
  }
  
  public long[] nextLong(int n) {
    long[] array = new long[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextLong();
    }
    return array;
  }
  
  public double[] nextDouble(int n) {
    double[] array = new double[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextDouble();
    }
    return array;
  }
  
  public char[] nextCharArray() {
    return next().toCharArray();
  }
  
  public int[][] nextInt(int n, int m) {
    int[][] matrix = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextInt();
      }
    }
    return matrix;
  }
  
  public char[][] nextChar(int n, int m) {
    char[][] matrix = new char[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextChar();
      }
    }
    return matrix;
  }
  
  public long[][] nextLong(int n, int m) {
    long[][] matrix = new long[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextLong();
      }
    }
    return matrix;
  }
  
  public double[][] nextDouble(int n, int m) {
    double[][] matrix = new double[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextDouble();
      }
    }
    return matrix;
  }
  
  public char[][] nextCharArray(int n) {
    char[][] matrix = new char[n][];
    for (int i = 0; i < n; i++) {
      matrix[i] = next().toCharArray();
    }
    return matrix;
  }
}

class ModCalculator {
  private final long mod;
  private final ModCombinationCache modCombinationCache;
  
  ModCalculator(long mod) {
    this.mod = mod;
    this.modCombinationCache = new ModCombinationCache();
  }
  
  public long add(long a, long b) {
    return (a + b) % mod;
  }
  
  public long sub(long a, long b) {
    return (a - b + mod) % mod;
  }
  
  public long mul(long a, long b) {
    a %= mod;
    b %= mod;
    return (a * b) % mod;
  }
  
  public long pow(long a, long b) {
    if (b == 0) {
      return 1;
    }
    long v = pow(mul(a, a), b / 2);
    if (b % 2 == 1) {
      return mul(v, a);
    } else {
      return v;
    }
  }
  
  public long inverse(long a) {
    return pow(a, mod - 2);
  }
  
  public long div(long a, long b) {
    return mul(a, inverse(b));
  }
  
  public long getF(int n) {
    return modCombinationCache.getF(n);
  }
  
  public long getP(int n, int r) {
    return modCombinationCache.getP(n, r);
  }
  
  public long getC(int n, int k) {
    return modCombinationCache.getC(n, k);
  }
  
  class ModCombinationCache {
    private final List<Long> factorialCache;
    private final List<Long> factorialInverseCache;
    
    public ModCombinationCache() {
      factorialCache = new ArrayList<>();
      factorialCache.add(1L);
      factorialInverseCache = new ArrayList<>();
      factorialInverseCache.add(1L);
    }
    
    private void resize(int n) {
      for (int i = factorialCache.size() - 1; i < n; i++) {
        long v = mul(factorialCache.get(i), i + 1);
        factorialCache.add(v);
        factorialInverseCache.add(inverse(v));
      }
    }
    
    long getF(int n) {
      resize(n);
      return factorialCache.get(n);
    }
    
    long getP(int n, int r) {
      resize(n);
      return mul(factorialCache.get(n), factorialInverseCache.get(n - r));
    }
    
    long getC(int n, int k) {
      resize(n);
      return mul(factorialCache.get(n), mul(factorialInverseCache.get(k), factorialInverseCache.get(n-k)));
    }
  }
}