F - path pass i

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.

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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.

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3
1 2 1
1 2
2 3

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.

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1
1

1

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2
1 2
1 2

2
2

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

5
8
10
5
5

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

18
15
0
14
23
0
23
0