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, fastout};
use proconio::marker::Usize1;
use std::collections::HashMap;
use std::mem::swap;

#[derive(Clone, Copy, Debug, Default)]
struct Adjacent {
    to: usize,
}

#[derive(Clone, Debug, Default)]
struct Graph {
    n: usize,
    links: Vec<Vec<Adjacent>>,
}

impl Graph {
    fn new(n: usize) -> Graph {
        Graph {
            n,
            links: vec![vec![]; n],
        }
    }

    fn add_edge(&mut self, from: usize, to: usize) {
        self.links[from].push(Adjacent { to });
        self.links[to].push(Adjacent { to: from });
    }
}

fn pair_count(n: usize) -> i64 { (n * (n + 1) / 2) as i64 }

fn get_or_default<'a, T, U>(hm: &'a HashMap<T, U>, key: &T, def: &'a U) -> &'a U
    where
        T: std::cmp::Eq + std::hash::Hash,
{
    match hm.get(key) {
        Some(result) => result,
        None => def,
    }
}

fn dfs(v: usize, p: usize, colors: &Vec<usize>, g: &Graph, ans: &mut Vec<i64>) -> (HashMap<usize, usize>, usize) {
    let mut unable_count = HashMap::<usize, usize>::new();
    let mut subtree_count = 1usize;
    let cv = &colors[v];

    for u in &g.links[v] {
        if u.to == p { continue; }
        let result = dfs(u.to, v, colors, g, ans);
        subtree_count += result.1;

        ans[*cv] -= pair_count(result.1 - *get_or_default(&result.0, cv, &0));

        let mut merge = result.0;
        if unable_count.len() < merge.len() {
            swap(&mut unable_count, &mut merge);
        }
        for (c, ubl) in merge.iter() {
            *unable_count.entry(*c).or_insert(0) += *ubl;
        }
    }

    *unable_count.entry(*cv).or_insert(0) = subtree_count;

    (unable_count, subtree_count)
}

#[fastout]
fn main() {
    input! {
        n: usize,
        ccc: [Usize1; n],
        ab: [(Usize1, Usize1); n - 1],
    }

    // println!("{:?} {:?} {:?}", n, ccc, ab);

    let mut g = Graph::new(n);
    for (a, b) in ab {
        g.add_edge(a, b);
    }

    let mut ans = vec![pair_count(n); n];

    let (unable_count, subtree_count) = dfs(0, 0, &ccc, &g, &mut ans);

    assert_eq!(n, subtree_count);

    for c in 0..n {
        println!("{:?}", ans[c] - pair_count(n - get_or_default(&unable_count, &c, &0)));
    }
}