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)

#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
#[allow(unused_imports)]
use std::cmp::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};
#[allow(unused_imports)]
use std::iter::FromIterator;
#[macro_export]
macro_rules ! dvec { ( $ t : expr ; $ len : expr ) => { vec ! [ $ t ; $ len ] } ; ( $ t : expr ; $ len : expr , $ ( $ rest : expr ) ,* ) => { vec ! [ dvec ! ( $ t ; $ ( $ rest ) ,* ) ; $ len ] } ; }
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[macro_export]
macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut parser = Parser :: from_str ( $ s ) ; input_inner ! { parser , $ ( $ r ) * } } ; ( parser = $ parser : ident , $ ( $ r : tt ) * ) => { input_inner ! { $ parser , $ ( $ r ) * } } ; ( new_stdin_parser = $ parser : ident , $ ( $ r : tt ) * ) => { let stdin = std :: io :: stdin ( ) ; let reader = std :: io :: BufReader :: new ( stdin . lock ( ) ) ; let mut $ parser = Parser :: new ( reader ) ; input_inner ! { $ parser , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { input ! { new_stdin_parser = parser , $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! input_inner { ( $ parser : ident ) => { } ; ( $ parser : ident , ) => { } ; ( $ parser : ident , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ parser , $ t ) ; input_inner ! { $ parser $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! read_value { ( $ parser : ident , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ parser , $ t ) ) ,* ) } ; ( $ parser : ident , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ parser , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ parser : ident , chars ) => { read_value ! ( $ parser , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ parser : ident , usize1 ) => { read_value ! ( $ parser , usize ) - 1 } ; ( $ parser : ident , $ t : ty ) => { $ parser . next ::<$ t > ( ) . expect ( "Parse error" ) } ; }
use std::io;
use std::io::BufRead;
use std::str;
pub struct Parser<R> {
    reader: R,
    buf: Vec<u8>,
    pos: usize,
}
impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}
impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                if len == 0 {
                    break;
                }
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return str::from_utf8(&self.buf[begin..end]).unwrap().parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
}
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
const BIG_STACK_SIZE: bool = true;
#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}
struct Rec {
    dp: Vec<u64>,
    g: Vec<Vec<usize>>,
    par: Vec<Option<usize>>,
    M: u64,
}
impl Rec {
    // vが黒か白かの場合の数
    fn solve(&mut self, v: usize) -> u64 {
        // 黒の場合
        let mut b = 1;
        for &u in &self.g[v].clone() {
            if let Some(p) = self.par[v] {
                if u == p { continue; }
            }
            b *= self.solve(u);
            b %= self.M;
        }
        // 白の場合のため+1
        let mut res = b+1;
        self.dp[v]=res%self.M;
        self.dp[v]
    }
}

static mut M: u64 = 0;
fn solve() {
    input!{
        N:usize,m:u64,
        xy:[(usize,usize);N-1]
    }
    unsafe {
        M = m;
    }
    let mut g = vec![vec![];N];
    for (x,y) in xy {
        g[x-1].push(y-1);
        g[y-1].push(x-1);
    }
    let mut reroot: ReRooting<F> = ReRooting::new(g);
    reroot.dfs(0);
    for u in 0..N {
        println!("{}", reroot.solve(u));
    }
}
struct F {}
impl Foldable for F {
    type T = u64;
    fn identity() -> u64 {
        1
    }
    fn fold(acc: &u64, x: &u64) -> u64 {
        unsafe {
            (acc * (x+1)) % M
        }
    }
    fn merge(a: &u64, b: &u64) -> u64 {
        unsafe {
            (a * b) % M
        }
    }
}
trait Foldable {
    type T: Clone + std::fmt::Debug;
    fn identity() -> Self::T;
    fn fold(acc: &Self::T, x: &Self::T) -> Self::T;
    fn merge(a: &Self::T, b: &Self::T) -> Self::T {
        let mut acc = Self::identity();
        acc = Self::fold(&acc, a);
        acc = Self::fold(&acc, b);
        acc
    }
}
struct ReRooting<F: Foldable> {
    g: Vec<Vec<usize>>,
    dp: Vec<Vec<F::T>>,
}
impl<F: Foldable> ReRooting<F> {
    fn new(g: Vec<Vec<usize>>) -> Self {
        let mut dp = vec![];
        for u in 0..g.len() {
            dp.push(vec![F::identity(); g[u].len()]);
        }
        ReRooting {
            g: g,
            dp: dp,
        }
    }

    fn dfs1(&mut self, par: Option<usize>, u: usize) -> F::T {
        let mut ret = F::identity();
        for i in 0..self.g[u].len() {
            let v = self.g[u][i];
            if Some(v) == par { continue; } 

            let r = self.dfs1(Some(u), v);
            self.dp[u][i] = r.clone();
            ret = F::fold(&ret, &r);
        }
        ret
    }

    fn dfs2(&mut self, par: Option<usize>, u: usize, u2par: F::T) {
        let n = self.g[u].len();
        let mut prefix = vec![F::identity(); n];
        let mut suffix = vec![F::identity(); n];
        for i in 0..n {
            let v = self.g[u][i];
            if Some(v) == par {
                self.dp[u][i] = u2par.clone();
            }
        }
        for i in 0..n {
            let l = if i>0 { prefix[i-1].clone() } else { F::identity() };
            prefix[i] = F::fold(&l, &self.dp[u][i]);
        }
        for i in (0..n).rev() {
            let r = if i < n-1 { suffix[i+1].clone() } else { F::identity() };
            suffix[i] = F::fold(&r, &self.dp[u][i]);
        }
        for i in 0..n {
            let v = self.g[u][i];
            if Some(v) == par { continue; }
            
            // 枝iを除いた値を計算する
            let l = if i>0 { prefix[i-1].clone() } else { F::identity() };
            let r = if i<n-1 { suffix[i+1].clone() } else { F::identity() };
            self.dfs2(Some(u), v, F::merge(&l, &r));
        }
    }

    pub fn dfs(&mut self, u: usize) {
        let p = self.dfs1(None, u);
        self.dfs2(None, u, p);
    }

    pub fn solve(&self, u: usize) -> F::T {
        let mut acc = F::identity();
        for d in &self.dp[u] {
            acc = F::fold(&acc, d);
        }
        acc
    }
}