Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices numbered 1 to N. The i-th edge in this tree connects Vertex a_i and b_i.
Additionally, each vertex is painted in a color, and the color of Vertex i is c_i. Here, the color of each vertex is represented by an integer between 1 and N (inclusive). The same integer corresponds to the same color; different integers correspond to different colors.

For each k=1, 2, ..., N, solve the following problem:

Find the number of simple paths that visit a vertex painted in the color k one or more times.

Note: The simple paths from Vertex u to v and from v to u are not distinguished.

Constraints

1 \leq N \leq 2 \times 10^5

1 \leq c_i \leq N

1 \leq a_i,b_i \leq N

The given graph is a tree.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
c_1 c_2 ... c_N
a_1 b_1
:
a_{N-1} b_{N-1}

Output

Print the answers for k = 1, 2, ..., N in order, each in its own line.

Sample Input 1

3
1 2 1
1 2
2 3

Sample Output 1

5
4
0

Let P_{i,j} denote the simple path connecting Vertex i and j.

There are 5 simple paths that visit a vertex painted in the color 1 one or more times:

P_{1,1}\,,\,
P_{1,2}\,,\,
P_{1,3}\,,\,
P_{2,3}\,,\,
P_{3,3}

There are 4 simple paths that visit a vertex painted in the color 2 one or more times:

P_{1,2}\,,\,
P_{1,3}\,,\,
P_{2,2}\,,\,
P_{2,3}

There are no simple paths that visit a vertex painted in the color 3 one or more times.

Sample Input 2

1
1

Sample Output 2

1

Sample Input 3

2
1 2
1 2

Sample Output 3

2
2

Sample Input 4

5
1 2 3 4 5
1 2
2 3
3 4
3 5

Sample Output 4

5
8
10
5
5

Sample Input 5

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

Sample Output 5

18
15
0
14
23
0
23
0

Solution in PyPy3 (7.3.0) (judge verdict: Wrong Answer)

from copy import deepcopy
n=int(input())
col=list(map(int,input().split()))
children=[[] for i in range(n)]
n_path=[0]*n
c_path=[[0]*n for i in range(n)]
checked=[False]*n
num_check=0
is_inv=False
for i in range(n):
  col[i]-=1
for i in range(n-1):
  a,b=map(int,input().split())
  children[a-1].append(b-1)
for i in children:
  for j in range(len(i)):
    if(checked[j]):
      is_inv=True
      break
    checked[j]=True
if(is_inv):
  tmp_children=deepcopy(children)
  children=[[] for i in range(n)]
  for i in range(n):
    for j in range(len(tmp_children[i])):
      children[tmp_children[i][j]].append(i)
checked=[False]*n
while num_check<n:
  for parent in range(n):
    if(checked[parent]):
      continue
    computable=True
    for child in children[parent]:
      if(not checked[child]):
        computable=False
        break
    if(not computable):
      continue
    checked[parent]=True
    num_check+=1
    for child in children[parent]:
      n_path[parent]+=n_path[child]
      for col_i in range(n):
        c_path[parent][col_i]+= (n_path[child] if col[parent]==col_i else c_path[child][col_i])
    n_path[parent]+=1
    c_path[parent][col[parent]]+=1
for col_i in range(n):
  ans=0
  for p in range(n):
    if(col[p]==col_i):
      ans+=n_path[p]
      for c1 in children[p]:
        for c2 in children[p]:
          if(c1>=c2):
            continue
          ans+=n_path[c1]*n_path[c2]
    else:
      ans+=c_path[p][col_i]
      for c1 in children[p]:
        for c2 in children[p]:
          if(c1==c2):
            continue
          ans+=c_path[c1][col_i]*(n_path[c2] if c2>c1 else n_path[c2]-c_path[c2][col_i])
  print(ans)