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: Time Limit Exceeded)

class Tree
  def initialize(n, colors, edges)
    @n = n
    @colors = colors
    @adj = @n.times.map{Array.new}
    edges.each do |a, b|
      @adj[a].push b
      @adj[b].push a
    end

    # 計算用
    @cut_color = nil
    @root_queue = []
  end

  def get_answer
    total_path_count = all_path_count(@n)
    (1..@n).map do |color|
      tree_sizes = get_tree_sizes_for_cut_color(color)
      path_count_in_trees = tree_sizes.map{|tree_size| all_path_count(tree_size)}.sum()
      total_path_count - path_count_in_trees
    end
  end

  private

  # colorのノードを取り除いて木を森にしたとき、
  # 森に含まれる木のサイズのリストを得る
  def get_tree_sizes_for_cut_color(color)
    tree_sizes = []
    @cut_color = color
    @root_queue.push [0, -1]
    until @root_queue.empty?
      root, root_parent = @root_queue.shift
      tree_size = get_subtree_size(root, root_parent)
      tree_sizes.push tree_size if tree_size > 0
    end
    tree_sizes
  end

  # node以下の部分木のサイズを返す
  # （nodeが@cut_colorの場合、そこで木は切れるので、
  # 子を新しい木のルートとして登録する）
  def get_subtree_size(node, parent)
    size = 0
    if @colors[node] == @cut_color
      @adj[node].each do |child|
        if child != parent
          @root_queue.push [child, node]
        end
      end
    else
      size = 1 # 自分自身
      @adj[node].each do |child|
        if child != parent
          size += get_subtree_size(child, node)
        end
      end
    end
    size
  end

  # サイズnの木にあるパスの総数
  def all_path_count(n)
    n * (n + 1) / 2
  end
end

n = gets.to_i
colors = gets.split.map(&:to_i)
edges = (n-1).times.map{ gets.split.map{|e| e.to_i - 1} }

tree = Tree.new(n, colors, edges)
tree.get_answer.each do |i|
  puts i
end