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;

type Graph = Vec<Vec<usize>>;

fn build_graph(n:usize,edges:Vec<(usize,usize)>) -> Graph{
    let mut g = vec![Vec::<usize>::new();n];
    for (a,b) in edges {
        g[a].push(b);
        g[b].push(a);
    }
    g
}

fn dfs_sz(v:usize,pre:usize,g:&Graph,mut sz:&mut Vec<i64>) -> i64{
    sz[v] = 1;
    for &to in &g[v] {
        if pre==to {
            continue
        }
        sz[v] += dfs_sz(to,v,&g,&mut sz);
    }
    sz[v]
}

fn dfs(v:usize,pre:usize,c:&Vec<usize>,g:&Graph,sz:&Vec<i64>,mut vst:&mut Vec<Vec<usize>>,mut buf:&mut Vec<i64>,mut rest:&mut Vec<i64>,mut ans:&mut Vec<i64>){
    vst[c[v]].push(v);
    let mut sum = 0i64;
    let mut sqsum = 0i64;
    for &to in &g[v] {
        if to==pre {
            continue;
        }
        let pre_buf = buf[v];
        dfs(to,v,&c,&g,&sz,&mut vst,&mut buf,&mut rest,&mut ans);
        let rsz = sz[to] - (buf[v] - pre_buf);
        sum+=rsz;
        sqsum+=rsz*rsz;
    }
    vst[c[v]].pop();

    rest[c[v]]-=sz[v]-buf[v];
    ans[c[v]]+=(sz[v]-buf[v])*(rest[c[v]]+1);
    ans[c[v]]+=(sum*sum-sqsum)/2;
    
    if !vst[c[v]].is_empty() {
        let par = *vst[c[v]].last().unwrap();
        buf[par] += sz[v];
    }

}

fn main() {
    let (n,g,c) = {
        input! {
            n:usize,
            mut c:[usize;n],
            mut edges:[(usize,usize);n-1]
        };
        c.iter_mut().for_each(|x|*x-=1);
        edges.iter_mut().for_each(|(a,b)|{*a-=1;*b-=1});
        let g = build_graph(n,edges);
        (n,g,c)
    };
    
    let mut sz = vec![0i64;n];
    dfs_sz(0,n,&g,&mut sz);

    let mut buf = vec![0i64;n];
    let mut rest = vec![n as i64;n];
    let mut ans = vec![0i64;n];
    let mut vst = vec![Vec::<usize>::new();n];
    dfs(0,n,&c,&g,&sz,&mut vst,&mut buf,&mut rest,&mut ans);
    
    
    for v in ans {
        println!("{}",v);
    }

}