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: Wrong Answer)

struct DFS {
    c: Vec<usize>,
    g: Vec<Vec<usize>>,
    v: Vec<i64>,
    res: Vec<i64>,
}

impl DFS {
    fn new(n: usize, c: Vec<usize>, g: Vec<Vec<usize>>) -> DFS {
        DFS {
            c, g,
            v: vec![0; n],
            res: vec![(n * (n + 1) / 2) as i64; n],
        }
    }

    fn search(&mut self, t: usize, u: usize) -> i64 {
        let mut k = 0;
        for v in self.g[u].clone() {
            if t == v { continue; }
            k += self.search(u, v);
            let x = &mut self.v[self.c[u]];
            self.res[self.c[u]] -= (*x + k) * (*x + k + 1) / 2;
            *x -= *x + k;
        }
        self.v[self.c[u]] -= 1;
        k + 1
    }
}

fn main () {
    let n: usize = {
        let mut buf = String::new();
        std::io::stdin().read_line(&mut buf).unwrap();
        buf.trim_end().parse().unwrap()
    };

    let c: Vec<usize> = {
        let mut buf = String::new();
        std::io::stdin().read_line(&mut buf).unwrap();
        let iter = buf.split_whitespace();
        iter.map(|x| x.parse::<usize>().unwrap() - 1).collect()
    };

    let mut g: Vec<Vec<usize>> = vec![vec![]; n];
    for _ in 0..(n - 1) {
        let (a, b): (usize, usize) = {
            let mut buf = String::new();
            std::io::stdin().read_line(&mut buf).unwrap();
            let mut iter = buf.split_whitespace();
            (
                iter.next().unwrap().parse::<usize>().unwrap() - 1,
                iter.next().unwrap().parse::<usize>().unwrap() - 1,
            )
        };
        g[a].push(b);
        g[b].push(a);
    }

    let mut dfs = DFS::new(n, c, g);
    dfs.search(!0, 0);
    for i in 0..n {
        dfs.res[i] -= (dfs.v[i] + n as i64) * (dfs.v[i] + n as i64 + 1) / 2
    }
    let ans = dfs.res.iter().map(|x| x.to_string()).collect::<Vec<_>>().join("\n");
    println!("{}", ans);
}