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::input;
use std::cmp;

fn dfs(g: &Vec<Vec<usize>>, c: &Vec<usize>, res: &mut Vec<usize>, cnt: &mut Vec<i64>,
       v: usize, p: i64) -> i64 {
  let mut sz: i64 = 1;
  let tmp = cnt[c[v]];

  for &u in &g[v] {
    cnt[c[v]] = 0;
    if u as i64 != p {
      sz += dfs(g, c, res, cnt, u, v as i64);
    }
  }

  cnt[c[v]] = -sz + tmp;

  if p == -1 {
    for (i, &x) in cnt.iter().enumerate() {
      let x = (x + sz) as usize;
      if x != 0 {
        res[i] -= x * (x - 1) / 2 + x;
      }
    }
  } else {
    let x = (cnt[c[p as usize]] + sz) as usize;
    if x != 0 {
      res[c[p as usize]] -= x * (x-1) / 2 + x;
      // println!("{} {}: {} {}", v, c[p as usize], x, res[c[p as usize]]);
    }
  }

  // for (i, &x) in cnt.iter().enumerate() {
    // print!("({}: {}) ", i, x);
  // }
  // println!();

  sz
}

fn main() {
  input! {
    n: usize,
    mut c: [usize; n],
    e: [(usize, usize); n-1],
  }
  let mut g = vec![Vec::<usize>::new(); n];
  for x in &mut c {
    *x -= 1;
  }
  for (a, b) in e {
    g[a-1].push(b-1);
    g[b-1].push(a-1);
  }

  let mut res = vec![n*(n-1)/2 + n; n];
  let mut cnt = vec![0i64; n];

  dfs(&g, &c, &mut res, &mut cnt, 0, -1);

  for x in res {
    println!("{}", x);
  }
}