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 std::collections::{BTreeMap, VecDeque};

fn main() {
    let (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());

    let n: usize = sc.read();
    let mut color = vec![];
    for _ in 0..n {
        color.push(sc.read::<usize>() - 1);
    }

    let mut graph = vec![vec![]; n];
    for _ in 1..n {
        let a = sc.read::<usize>() - 1;
        let b = sc.read::<usize>() - 1;
        graph[a].push(b);
        graph[b].push(a);
    }

    let mut subtree_size = vec![0; n];
    let mut color_graph = vec![vec![]; n];
    dfs(
        0,
        0,
        &graph,
        &color,
        &mut color_graph,
        &mut subtree_size,
        &mut vec![vec![]; n],
        &mut vec![vec![]; n],
    );

    let mut by_color = vec![vec![]; n];
    for i in 0..n {
        by_color[color[i]].push(i);
    }

    for c in 0..n {
        let vertices = &by_color[c];
        let mut count = 0;
        let mut sub_size = 0;
        for &v in vertices.iter() {
            sub_size += subtree_size[v];
            for a in color_graph[v].iter() {
                let next = a.0;
                let mut s = subtree_size[next];
                for &child in a.1.iter() {
                    s -= subtree_size[child];
                    sub_size -= subtree_size[child];
                }

                if s > 0 {
                    count += s * (s - 1) / 2;
                }
            }
        }

        let remain = n - sub_size;
        if remain > 0 {
            count += remain * (remain - 1) / 2;
        }

        let ans = n * (n - 1) / 2 + vertices.len() - count;
        println!("{}", ans);
    }
}

fn dfs(
    v: usize,
    p: usize,
    graph: &Vec<Vec<usize>>,
    color: &Vec<usize>,
    color_graph: &mut Vec<Vec<(usize, Vec<usize>)>>,
    subtree_size: &mut Vec<usize>,
    color_stack: &mut Vec<Vec<usize>>,
    parent_stack: &mut Vec<Vec<usize>>,
) {
    let v_color = color[v];
    if let Some(parent) = color_stack[v_color].last().cloned() {
        parent_stack[parent].push(v);
    }

    color_stack[v_color].push(v);

    for &next in graph[v].iter() {
        if p == next {
            continue;
        }
        dfs(
            next,
            v,
            graph,
            color,
            color_graph,
            subtree_size,
            color_stack,
            parent_stack,
        );
        subtree_size[v] += subtree_size[next];
        color_graph[v].push((next, parent_stack[v].drain(0..).collect()));
        assert_eq!(parent_stack[v].len(), 0);
    }

    subtree_size[v] += 1;
    assert_eq!(color_stack[v_color].pop(), Some(v));
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> Self {
        Self(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: ToString>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.to_string().as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}