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)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
#[allow(unused_imports)]
use itertools::Itertools;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;

fn dfs(t: &Vec<Vec<usize>>, res: &mut Vec<(usize, usize)>, v: usize, p: usize, k: &mut usize) {
    res[v].0 = *k;
    *k += 1;
    for &u in &t[v] {
        if u == p {
            continue;
        }
        dfs(t, res, u, v, k);
    }
    res[v].1 = *k;
}

fn main() {
    input! {
        n: usize,
        c: [Usize1; n],
        ab: [(Usize1, Usize1); n - 1],
    }
    let mut t = vec![vec![]; n];
    for (a, b) in ab {
        t[a].push(b);
        t[b].push(a);
    }

    let f = |x| x * (x + 1) / 2;
    let mut io = vec![(0, 0); n];
    dfs(&t, &mut io, 0, 0, &mut 0);

    let mut clist = vec![vec![]; n];
    for i in 0..n {
        clist[c[i]].push(i);
    }
    let mut bit = kyopro_lib::bit::BIT::new(n);
    for i in 0..n {
        bit.add(i, 1);
    }
    for mut c in clist {
        let mut res = f(n as i64);
        c.sort_by_key(|i| std::cmp::Reverse(io[*i].0));
        let mut hist = vec![];
        for v in c {
            let mut cnt = 1;
            for &u in &t[v] {
                if io[u].0 < io[v].0 {
                    continue;
                }
                let x = bit.sum_range(io[u].0..io[u].1);
                cnt += x;
                res -= f(x);
            }
            bit.add(io[v].0, -cnt);
            hist.push((io[v].0, cnt));
        }
        println!("{}", res - f(bit.sum_range(0..n)));
        for (v, c) in hist {
            bit.add(v, c);
        }
    }
}
*/

fn main() {
    let exe = "/tmp/binC2E38E15";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &str) -> Vec<u8> {
    let mut ret = vec![];
    let mut buf = 0;
    let mut tbl = vec![64; 256];
    for i in 0..64 { tbl[TBL[i] as usize] = i as u8; }
    for (i, c) in v.bytes().filter_map(|c| { let c = tbl[c as usize]; if c < 64 { Some(c) } else { None } }).enumerate() {
        match i % 4 {
            0 => buf = c << 2,
            1 => { ret.push(buf | c >> 4); buf = c << 4; }
            2 => { ret.push(buf | c >> 2); buf = c << 6; }
            3 => ret.push(buf | c),
            _ => unreachable!(),
        }
    }
    ret
}

const TBL: &'static [u8] = b"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
const BIN: &'static str = "<base64-encoded ELF executable removed>";