Problem:
Score : 100 points

Problem Statement

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.

Sample Input 1

3 100
1 2
2 3

Sample Output 1

3
4
3

There are seven ways to paint the vertices, as shown in the figure below.
Among them, there are three ways such that Vertex 1 is black, four ways such that Vertex 2 is black and three ways such that Vertex 3 is black.

Sample Input 2

4 100
1 2
1 3
1 4

Sample Output 2

8
5
5
5

Sample Input 3

1 100

Sample Output 3

1

Sample Input 4

10 2
8 5
10 8
6 5
1 5
4 8
2 10
3 6
9 2
1 7

Sample Output 4

0
0
1
1
1
0
1
0
1
1

Be sure to print the answers modulo M.

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[allow(unused_imports)]
use std::cmp;

// Input macro from https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
  (source = $s:expr, $($r:tt)*) => {
    let mut iter = $s.split_whitespace();
    input_inner!{iter, $($r)*}
  };
  ($($r:tt)*) => {
    #[allow(unused_mut)]
    let mut s = {
      use std::io::Read;
      let mut s = String::new();
      std::io::stdin().read_to_string(&mut s).unwrap();
      s
    };
    let mut iter = s.split_whitespace();
    input_inner!{iter, $($r)*}
  };
}

macro_rules! input_inner {
  ($iter:expr) => {};
  ($iter:expr, ) => {};

  ($iter:expr, $var:ident : $t:tt $($r:tt)*) => {
    let $var = read_value!($iter, $t);
    input_inner!{$iter $($r)*}
  };
}

macro_rules! read_value {
  ($iter:expr, ( $($t:tt),* )) => {
    ( $(read_value!($iter, $t)),* )
  };

  ($iter:expr, [ $t:tt ; $len:expr ]) => {
    (0..$len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
  };

  ($iter:expr, chars) => {
    read_value!($iter, String).chars().collect::<Vec<char>>()
  };

  ($iter:expr, usize1) => {
    read_value!($iter, usize) - 1
  };

  ($iter:expr, $t:ty) => {
    $iter.next().unwrap().parse::<$t>().expect("Parse error")
  };
}

fn bottom_up(vs: &Vec<Vec<usize>>, sub_pattern: &mut Vec<i64>, children: &mut Vec<Vec<(usize, i64)>>, v: usize, m: i64) -> i64 {
  let mut pat = 1;
  sub_pattern[v] = 1;
  for &u in vs[v].iter() {
    if sub_pattern[u] >= 0 {
      continue;
    }
    let p = bottom_up(vs, sub_pattern, children, u, m);
    pat = (pat * (p + 1)) % m;
    children[v].push((u, p));
  }
  sub_pattern[v] = pat;
  pat
}

fn top_down(sub_pattern: &Vec<i64>, children: &Vec<Vec<(usize, i64)>>, pattern: &mut Vec<i64>, par_pattern: i64, v: usize, m: i64) {
  pattern[v] = (par_pattern + 1) * sub_pattern[v] % m;
  let mut left_scan = vec![1i64; children[v].len() + 1];
  for (i, &(_, p)) in children[v].iter().enumerate() {
    left_scan[i+1] = left_scan[i] * (p + 1) % m;
  }
  let mut right_scan = vec![1i64; children[v].len() + 1];
  for (i, &(_, p)) in children[v].iter().enumerate().rev() {
    right_scan[i] = right_scan[i+1] * (p + 1) % m;
  }
  for (i, &(u, _)) in children[v].iter().enumerate() {
    let pp = left_scan[i] * right_scan[i+1] % m * (par_pattern + 1) % m;
    top_down(sub_pattern, children, pattern, pp, u, m);
  }
}

fn main() {
  input! {
    n: usize,
    m: i64,
    xys: [(usize1, usize1); n-1]
  }
  let mut vs = vec![vec![]; n];
  for &(x, y) in xys.iter() {
    vs[x].push(y);
    vs[y].push(x);
  }
  let mut sub_pattern = vec![-1i64; n];
  let mut children = vec![vec![]; n];
  bottom_up(&vs, &mut sub_pattern, &mut children, 0, m);
  let mut pattern = vec![0i64; n];
  top_down(&sub_pattern, &children, &mut pattern, 0, 0, m);
  for &p in pattern.iter() {
    println!("{}", p);
  }
}