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 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

//treeDP-allside
//https://atcoder.jp/contests/dp/tasks/dp_v
//https://ei1333.hateblo.jp/entry/2017/04/10/224413
//https://lumakernel.github.io/ecasdqina/graph/DP-all-subtree
#include<stdio.h>
#include<string.h>
#include<stdlib.h>
#include<stdbool.h>
#include<time.h>
#include<assert.h>
#define inf 1072114514
#define llinf 4154118101919364364
#define mod 1000000007
#define pi 3.1415926535897932384

int max(int a,int b){if(a>b){return a;}return b;}
int min(int a,int b){if(a<b){return a;}return b;}
int zt(int a,int b){return max(a,b)-min(a,b);}
int round(int a,int b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
int ceil(int a,int b){if(a%b==0){return a/b;}return (a/b)+1;}
int gcd(int a,int b){int c;while(b!=0){c=a%b;a=b;b=c;}return a;}
int lcm(int a,int b){int c=gcd(a,b);a/=c;return a*b;}
int nCr(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
int nHr(int a,int b){return nCr(a+b-1,b);}
int fact(int a){int i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
int pow(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
int dsum(int x){int r=0;while(x){r+=(x%10);x/=10;}return r;}
int dsumb(int x,int b){int r=0;while(x){r+=(x%b);x/=b;}return r;}
int sankaku(int x){return ((1+x)*x)/2;}
void swap(int *a,int *b){int c;c=(*a);(*a)=(*b);(*b)=c;}
long long llmax(long long a,long long b){if(a>b){return a;}return b;}
long long llmin(long long a,long long b){if(a<b){return a;}return b;}
long long llzt(long long a,long long b){return llmax(a,b)-llmin(a,b);}
long long llround(long long a,long long b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
long long llceil(long long a,long long b){if(a%b==0){return a/b;}return (a/b)+1;}
long long llgcd(long long a,long long b){long long c;while(b!=0){c=a%b;a=b;b=c;}return a;}
long long lllcm(long long a,long long b){long long c=llgcd(a,b);a/=c;return a*b;}
long long llnCr(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
long long llnHr(long long a,long long b){return llnCr(a+b-1,b);}
long long llfact(long long a){long long i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
long long llpow(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
long long lldsum(long long x){long long r=0;while(x){r+=(x%10);x/=10;}return r;}
long long lldsumb(long long x,long long b){long long r=0;while(x){r+=(x%b);x/=b;}return r;}
long long llsankaku(long long x){return ((1+x)*x)/2;}
void llswap(long long *a,long long *b){long long c;c=(*a);(*a)=(*b);(*b)=c;}
double dbmax(double a,double b){if(a>b){return a;}return b;}
double dbmin(double a,double b){if(a<b){return a;}return b;}
double dbzt(double a,double b){return dbmax(a,b)-dbmin(a,b);}
void dbswap(double *a,double *b){double c;c=(*a);(*a)=(*b);(*b)=c;}
void chswap(char *a,char *b){char c;c=(*a);(*a)=(*b);(*b)=c;}
int sortfncsj(const void *a,const void *b){if(*(int *)a>*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
int sortfnckj(const void *a,const void *b){if(*(int *)a<*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
int llsortfncsj(const void *a,const void *b){if(*(long long *)a>*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
int llsortfnckj(const void *a,const void *b){if(*(long long *)a<*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
int dbsortfncsj(const void *a,const void *b){if(*(double *)a>*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
int dbsortfnckj(const void *a,const void *b){if(*(double *)a<*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
int strsortfncsj(const void *a,const void *b){return strcmp((char *)a,(char *)b);}
int strsortfnckj(const void *a,const void *b){return strcmp((char *)b,(char *)a);}
int chsortfncsj(const void *a,const void *b){if(*(char *)a>*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
int chsortfnckj(const void *a,const void *b){if(*(char *)a<*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}

void shuffledget(int x[],int n){
    int i,b[524288],p,c;
    for(i=0;i<n;i++){
        b[i]=i;
    }
    for(i=n;i>=1;i--){
        p=rand()%i;
        c=b[i-1];b[i-1]=b[p];b[p]=c;
    }
    for(i=0;i<n;i++){
        scanf("%d",&x[b[i]]);
    }
}

int dx4[4]={1,-1,0,0};
int dy4[4]={0,0,1,-1};
int dx8[8]={-1,-1,-1,0,0,1,1,1};
int dy8[8]={-1,0,1,-1,1,-1,0,1};

int search(int x,int a[],int n){
    int st=0,fi=n-1,te;
    while(st<=fi){
        te=(st+fi)/2;
        if(a[te]<x){st=te+1;}else{fi=te-1;}
    }
    return st;
}

void prarr(int arr[],int n){
  int i;
  for(i=0;i<n;i++){
    if(i){printf(" ");}
    printf("%d",arr[i]);
  }
  printf("\n");
  return;
}

void getperm(int a[],int n){
  int i,p;
  for(i=0;i<n;i++){
    a[i]=i;
  }
  for(i=n-1;i>=1;i--){
    p=rand()%(i+1);
    swap(&a[p],&a[i]);
  }
}

typedef struct{
int val;
int node;
}sd;

int sdsortfnc(const void *a,const void *b){
if(((sd*)a)->val < ((sd*)b)->val){return -1;}
if(((sd*)a)->val > ((sd*)b)->val){return 1;}
return 0;
}

void coordinate_comp(int a[],int n){
  int i,c=0;
  sd dat[524288];
  for(i=0;i<n;i++){
    dat[i].val=a[i];
    dat[i].node=i;
  }
  qsort(dat,n,sizeof(dat[0]),sdsortfnc);
  a[dat[0].node]=c;
  for(i=1;i<n;i++){
    if(dat[i-1].val!=dat[i].val){c++;}
    a[dat[i].node]=c;
  }
}

typedef struct{
    long long st;
    long long fi;
    long long kr;
    long long id;
    long long rev;
}rs;

typedef struct{
    long long st;
    long long kz;
}mkj;

int sortfnc(const void *a,const void *b){
if(((rs*)a)->st > ((rs*)b)->st){return 1;}
if(((rs*)a)->st < ((rs*)b)->st){return -1;}
if(((rs*)a)->fi > ((rs*)b)->fi){return 1;}
if(((rs*)a)->fi < ((rs*)b)->fi){return -1;}
return 0;
}

int revsortfnc(const void *a,const void *b){
long long p,q,r,s,t;
p=((rs*)a)->st;
q=((rs*)a)->fi;
r=((rs*)b)->st;
s=((rs*)b)->fi;
if(p>q){t=p;p=q;q=t;}
if(r>s){t=r;r=s;s=t;}
if(p<r){return -1;}
if(p>r){return 1;}
if(q<s){return -1;}
if(q>s){return 1;}
return 0;
}

void makemkj(rs g[],mkj x[],long long n){
    long long i,ms=0,nst=g[0].st;
    for(i=1;i<n;i++){
        if(g[i].st!=g[i-1].st){
            x[nst].kz=i-ms;
            x[nst].st=ms;
            nst=g[i].st;ms=i;
        }
    }
    x[nst].kz=n-ms;
    x[nst].st=ms;
}

long long inmod;

long long calcfl[524288]={0};
long long dp[524288];

//repdp(v,-1,g,x) := pattern(v is a painted node)
long long repdp(long long v,long long bk,rs g[],mkj x[]){
  long long i,p,nr=1,ne,nrev,mem;
  long long *sta,*bksta;
  if(bk==-1){
    sta=malloc(sizeof(long long)*x[v].kz);
    bksta=malloc(sizeof(long long)*x[v].kz);
  }

  //DP getsubtree
  for(i=x[v].st;i<x[v].st+x[v].kz;i++){
    if(g[i].fi==bk){mem=g[i].id;continue;}
    p=i-x[v].st;
    ne=g[i].rev;
    if(calcfl[ne]==0){
      repdp(g[i].fi,v,g,x);
    }
    if(bk==-1){
      sta[p]=(dp[ne]+1)%inmod;
      bksta[p]=(dp[ne]+1)%inmod;
    }
    nr*=(dp[ne]+1);nr%=inmod;
  }

  //DP spread
  if(bk!=-1){
    dp[mem]=nr;
    calcfl[mem]=1;
    //printf("%lld -> %lld : %lld\n",g[mem].st,g[mem].fi,dp[mem]);
  }
  else{
    for(i=1;i<x[v].kz;i++){
      sta[i]*=sta[i-1];
      sta[i]%=inmod;
    }
    for(i=x[v].kz-2;i>=0;i--){
      bksta[i]*=bksta[i+1];
      bksta[i]%=inmod;
    }
    for(i=x[v].st;i<x[v].st+x[v].kz;i++){
      p=i-x[v].st;
      ne=g[i].id;
      dp[ne]=1;
      if(p!=0){dp[ne]*=sta[p-1];dp[ne]%=inmod;}
      if(p!=x[v].kz-1){dp[ne]*=bksta[p+1];dp[ne]%=inmod;}
      calcfl[ne]=1;
      //printf("%lld -> %lld : %lld\n",g[ne].st,g[ne].fi,dp[ne]);
    }
  }

  return nr;
}

int main(void){
  long long i,j,n,m,k,a,b,c,h,w,r=0,l,t,cnt=0;
  rs g[524288];
  mkj x[524288];
  scanf("%lld%lld",&n,&inmod);
  for(i=0;i<(n-1);i++){
    scanf("%lld%lld",&g[cnt].st,&g[cnt].fi);
    g[cnt].kr=1;
    cnt++;
    g[cnt]=g[cnt-1];
    llswap(&g[cnt].st,&g[cnt].fi);
    cnt++;
  }
  qsort(g,cnt,sizeof(g[0]),sortfnc);
  for(i=0;i<cnt;i++){
    g[i].id=i;
  }
  qsort(g,cnt,sizeof(g[0]),revsortfnc);
  for(i=0;i<cnt;i+=2){
    g[i].rev=g[i+1].id;
    g[i+1].rev=g[i].id;
  }
  qsort(g,cnt,sizeof(g[0]),sortfnc);
  makemkj(g,x,cnt);

  for(i=1;i<=n;i++){
    printf("%lld\n",repdp(i,-1,g,x));
  }
  return 0;
}