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 Rust (1.42.0) (judge verdict: Accepted)

use proconio::marker::Usize1;
fn main() {
    proconio::input! {
        n: usize,
        cs: [Usize1; n],
        ab: [(Usize1,Usize1); n-1],
    }
    let mut g = vec![vec![];n];
    for (a,b) in ab {
        g[a].push(b);
        g[b].push(a);
    }
    let mut all=0;
    let mut cnt = vec![0;n];
    let mut ans = vec![s(n as i64);n];
    f(0, 0, &cs, &g, &mut all, &mut cnt, &mut ans);
    for c in 0..n {
        ans[c] -= s(n as i64-cnt[c]);
    }
    for c in cs {
        ans[c] += 1;
    }
    for a in ans {
        println!("{}",a);
    }
}
fn f(v: usize, p: usize, cs: &Vec<usize>, g: &Vec<Vec<usize>>, all: &mut i64, cnt: &mut Vec<i64>, ans: &mut Vec<i64>)->i64 {
    let c = cs[v];
    let mut sub = 1;
    let bfr = cnt[c];
    for &u in &g[v] {
        if u==p {continue;}
        let b = *all-cnt[c];
        sub += f(u, v, cs, g, all, cnt, ans);
        ans[c] -= s(*all-cnt[c]-b)
    }
    cnt[c] = bfr+sub;
    *all+=1;
    sub
}
fn s(n: i64)->i64 {
    n*(n-1)/2
}