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)

struct DFS {
    c: Vec<usize>,
    g: Vec<Vec<usize>>,
    extra_size: Vec<i64>,
    ans: Vec<i64>,
}

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

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

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 {
        let x = n as i64 - dfs.extra_size[i];
        dfs.ans[i] -= x * (x + 1) / 2
    }

    println!(
        "{}",
        dfs.ans.iter()
            .map(|x| x.to_string())
            .collect::<Vec<_>>()
            .join("\n")
    );
}