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 Ruby (2.7.1) (judge verdict: Accepted)

N = gets.to_i
Cs = [nil] + gets.split.map(&:to_i)
ABs = readlines.map{|l| l.split.map(&:to_i)}

@g = ABs.each_with_object(Array.new(N + 1){[]}){|(a, b), h| h[a] << b; h[b] << a}
@m = Array.new(N + 1, 0)
@s = ->(n){n*(n + 1)/2}

def dfs(u, p)
  s = 1
  ur = Hash.new(0)
  c = Cs[u]
  @g[u].reject{_1 == p}.each do |v|
    s1, ur1 = dfs(v, u)
    s += s1
    @m[c] += @s[s1 - ur1[c]]
    ur, ur1 = ur1, ur if ur1.size > ur.size
    ur1.each{ur[_1] += _2}
  end
  ur[c] = s
  [s, ur]
end

_, ur = dfs(1, 0)
sn = @s[N]
(1 .. N).each{|c| puts sn - @m[c] - @s[N - ur[c]]}