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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
#define int long long
#define rep(i,n) for(int (i)=0;(i)<(n);(i)++)
#define rrep(i,n) for(int (i)=((n)-1);(i)>=0;(i)--)
#define itn int
#define miele(v) min_element(v.begin(), v.end())
#define maele(v) max_element(v.begin(), v.end())
#define SUM(v) accumulate(v.begin(), v.end(), 0LL)
#define lb(a, key) lower_bound(a.begin(),a.end(),key)
#define ub(a, key) upper_bound(a.begin(),a.end(),key)
#define COUNT(a, key) count(a.begin(), a.end(), key) 
#define BITCOUNT(x) __builtin_popcount(x)
#define pb push_back
#define all(x) (x).begin(),(x).end()
#define F first
#define S second
using P = pair <int,int>;
using WeightedGraph = vector<vector <P>>;
using UnWeightedGraph = vector<vector<int>>;
const long long INF = 1LL << 60;
const int MOD = 1000000007;
void YN(bool flg) {cout<<(flg?"YES":"NO")<<'\n';}
void Yn(bool flg) {cout<<(flg?"Yes":"No")<<'\n';}
void yn(bool flg) {cout<<(flg?"yes":"no")<<'\n';}
template <typename T> 
int getIndexOfLowerBound(vector <T> &v, T x){
    return lower_bound(v.begin(),v.end(),x)-v.begin();
}
template <typename T> 
int getIndexOfUpperBound(vector <T> &v, T x){
    return upper_bound(v.begin(),v.end(),x)-v.begin();
}
template<class T> inline bool chmin(T& a, T b) { if (a > b) { a = b; return true; } return false; }
template<class T> inline bool chmax(T& a, T b) { if (a < b) { a = b; return true; } return false; }
#define DUMPOUT cerr
#define repi(itr, ds) for (auto itr = ds.begin(); itr != ds.end(); itr++)
template <typename T, typename U>
istream &operator>>(istream &is, pair<T,U> &p_var) {
    is >> p_var.first >> p_var.second;
    return is;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &vec) {
    for (T &x : vec) is >> x;
    return is;
}
template <typename T>
istream &operator>>(istream &is, vector<vector<T>> &df) {
    for(int i=0;i<df.size();i++)for(int j=0;j<df[i].size();j++)
    is >> df[i][j];
    return is;
}
template <typename T, typename U>
ostream &operator<<(ostream &os, pair<T, U> &pair_var) {
    DUMPOUT<<'{';
    os << pair_var.first;
    DUMPOUT<<',';
    os << " "<< pair_var.second;
    DUMPOUT<<'}';
    return os;
}
template <typename T>
ostream &operator<<(ostream &os, vector<T> &vec) {
    DUMPOUT<<'[';
    for (int i = 0; i < vec.size(); i++) 
    os << vec[i] << (i + 1 == vec.size() ? "" : " ");
    DUMPOUT<<']';
    return os;
}
template <typename T>
ostream &operator<<(ostream &os, vector<vector<T>> &df) {
  for (auto& vec : df) os<<vec;
  return os;
}
template <typename T, typename U>
ostream &operator<<(ostream &os, map<T, U> &map_var) {
    DUMPOUT << "{";
    repi(itr, map_var) {
        os << *itr;
        itr++;
        if (itr != map_var.end()) DUMPOUT << ", ";
        itr--;
    }
    DUMPOUT << "}";
    return os;
}
template <typename T>
ostream &operator<<(ostream &os, set<T> &set_var) {
    DUMPOUT << "{";
    repi(itr, set_var) {
        os << *itr;
        itr++;
        if (itr != set_var.end()) DUMPOUT << ", ";
        itr--;
    }
    DUMPOUT << "}";
    return os;
}
void print() {cout << endl;}
template <class Head, class... Tail>
void print(Head&& head, Tail&&... tail) {
  cout << head;
  if (sizeof...(tail) != 0) cout << " ";
  print(forward<Tail>(tail)...);
}
void dump_func() {DUMPOUT << '#'<<endl;}
template <typename Head, typename... Tail>
void dump_func(Head &&head, Tail &&... tail) {
    DUMPOUT << head;
    if (sizeof...(Tail) > 0) DUMPOUT << ", ";
    dump_func(std::move(tail)...);
}
#ifdef DEBUG_
#define DEB
#define dump(...)                                                              \
    DUMPOUT << "  " << string(#__VA_ARGS__) << ": "                            \
            << "[" << to_string(__LINE__) << ":" << __FUNCTION__ << "]"        \
            << endl                                                            \
            << "    ",                                                         \
        dump_func(__VA_ARGS__)
#else
#define DEB if (false)
#define dump(...)
#endif
int n, m;
UnWeightedGraph g;
vector <int> data;
vector <vector<int>> l, r;
int dfs(int now = 0, int pre = -1){
    int res = 1;
    int x = 0;
    vector <int> tmp;

    for(auto e: g[now]){
        if(e != pre) {
            int multiply = dfs(e, now)+1;
            res *= multiply;
        }
        res %= m;
    }
    return data[now] = res;
}
vector <int> ans;
signed main(void) { cin.tie(0); ios::sync_with_stdio(false);
    cin>>n>>m;
    g.resize(n), data.resize(n), l.resize(n), r.resize(n), ans.resize(n);
    rep(i, n-1){
        int a,b; cin>>a>>b;
        a--; b--;
        g[a].pb(b);
        g[b].pb(a);
    }
    dfs();
    queue <tuple <int,int,int>> q;
    q.push(make_tuple(0, -1, -1));
    while(!q.empty()){
        int now = get <0>(q.front()), pre = get <1>(q.front()), idx = get <2>(q.front()); 
        q.pop();
        if(now == 0) ans[0] = data[0];
        l[now].pb(1);
        int x = 0;
        for(auto e:g[now]){
            if(e == pre){
                int multiply = l[pre][idx]*r[pre][idx+1]+1;
                multiply %= m;
                l[now].pb(multiply), r[now].pb(multiply);
            }else{
                l[now].pb(data[e]+1);
                r[now].pb(data[e]+1);
                q.push(make_tuple(e, now, x));
            }
            x++;
        }
        r[now].pb(1);
        for(int i=1;i<l[now].size();i++){
            l[now][i] *= l[now][i-1];
            l[now][i] %= m;
        }
        for(int i=l[now].size()-2;i>=0;i--){
            r[now][i] *= r[now][i+1];
            r[now][i] %= m;
        }
        if(now == 0) ans[0] = data[0];
        ans[now] = r[now][0];
    }
    rep(i, n) print(ans[i]);
}