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 Python (3.4.3) (judge verdict: Accepted)

class ReRooting(object):
    """ 全方位木DP """
    def processing(self, in_:list, v:int):
        """ 単項演算(加工) f """
        return [in_[0] + 1]
    
    def merge(self, in1:list, in2:list):
        """ 二項演算(マージ) ・ """
        return [in1[0] * in2[0] % M]

    def adjustment(self, in_:list, v:int):
        """ 単項演算(調整) g """
        return in_
    
    def solve(self):
        """
        計算結果を返す
        
        Ex. [cnt for cnt, _ in self.calc()]
        """
        return [res for [res] in self.calc()]

    def __init__(self, V:int, e:list):
        """
        V: 頂点数
        e: 単位元
        """
        self.V, self.e = V, e
        self.edge = [[] for _ in range(V)]
        self.par = [-1] * V
        self.order = []
    
    def add_edge(self, u:int, v:int):
        """ 辺(u,v)をグラフに追加する """
        self.edge[u].append(v)
        self.edge[v].append(u)
    
    def topological_sort(self, root:int = 0):
        """ トポロジカルソート """
        from collections import deque
        que = deque([root])
        while que:
            v = que.popleft()
            self.order.append(v)
            for w in self.edge[v]:
                if w != self.par[v]:
                    self.par[w] = v
                    self.edge[w].remove(v)
                    que.append(w)
    
    def bottom_up(self):
        """ Bottom-Up DP """
        accBU = [self.e for _ in range(self.V)]
        resBU = [None] * self.V
        for v in reversed(self.order):
            for c in self.edge[v]:
                accBU[v] = self.merge(accBU[v], self.processing(resBU[c], c))
            resBU[v] = self.adjustment(accBU[v], v)
        return accBU, resBU
    
    def top_down(self, resBU:list):
        """ Top-Down DP """
        accTD = [self.e for _ in range(self.V)]
        resTD = [self.e for _ in range(self.V)]

        for p in self.order:
            ac = self.e if (p == self.order[0]) else self.processing(resTD[p], p)
            for v in self.edge[p]:
                accTD[v] = ac[:]
                ac = self.merge(ac, self.processing(resBU[v], v))
            
            ac = self.e
            for v in reversed(self.edge[p]):
                accTD[v] = self.merge(accTD[v], ac)
                resTD[v] = self.adjustment(accTD[v], v)
                ac = self.merge(ac, self.processing(resBU[v], v))
        return resTD
    
    def calc(self):
        self.topological_sort()
        accBU, resBU = self.bottom_up()
        resTD = self.top_down(resBU)

        res = [None] * self.V
        for v in self.order:
            res[v] = self.e if (v == self.order[0]) else self.processing(resTD[v], v)
            res[v] = self.merge(accBU[v], res[v])
            res[v] = self.adjustment(res[v], v)
        return res

#############################################################
N, M = map(int, input().split())
 
RR = ReRooting(N, [1])
for _ in range(N - 1):
    x, y = map(int, input().split())
    RR.add_edge(x - 1, y - 1)
 
print(*RR.solve(), sep="\n")