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)

#![allow(dead_code)]
use std::cmp;
use std::collections::*;

pub struct Scanner {
    line: Vec<String>,
    idx: usize,
}

impl Scanner {
    pub fn new() -> Self {
        let mut buffer: Vec<u8> = Vec::new();
        std::io::Read::read_to_end(&mut std::io::stdin(), &mut buffer).unwrap();
        let b: &str = unsafe { std::str::from_utf8_unchecked(&mut buffer) };
        Scanner {
            line: b
                .trim()
                .split_ascii_whitespace()
                .collect::<Vec<&str>>()
                .iter()
                .map(|&s| String::from(s))
                .collect(),
            idx: 0,
        }
    }
    pub fn next<T>(&mut self) -> T
    where
        T: std::str::FromStr,
        T::Err: std::fmt::Debug,
    {
        let ret: T = self.line[self.idx].parse::<T>().unwrap();
        self.idx += 1;
        ret
    }
    pub fn next_vec<T>(&mut self, n: usize) -> Vec<T>
    where
        T: std::str::FromStr,
        T::Err: std::fmt::Debug,
    {
        let mut v: Vec<T> = Vec::with_capacity(n);
        for _i in 0..n {
            v.push(self.next());
        }
        v
    }
    pub fn next_chars(&mut self) -> Vec<char> {
        let s: String = self.next();
        s.chars().collect::<Vec<char>>()
    }
}

fn lower_bound<T: std::cmp::Ord>(v: &Vec<T>, val: &T) -> usize {
    let mut lb: isize = -1;
    let mut ub: isize = v.len() as isize;
    while ub - lb > 1 {
        let mid = (lb + ub) / 2;
        if *val <= v[mid as usize] {
            ub = mid;
        } else {
            lb = mid;
        }
    }
    return ub as usize;
}

fn dfs(
    u: usize,
    p: usize,
    dp: &mut Vec<i32>,
    ans: &mut Vec<usize>,
    a: &Vec<i32>,
    edge: &Vec<Vec<usize>>,
) {
    let idx = lower_bound(dp, &a[u]);
    let prev = dp[idx];
    dp[idx] = a[u];
    ans[u] = lower_bound(dp, &i32::max_value());

    for v in &edge[u] {
        if *v == p {
            continue;
        }
        dfs(*v, u, dp, ans, a, edge);
    }
    dp[idx] = prev;
}

fn main() {
    let mut scan = Scanner::new();
    let n: usize = scan.next();
    let a: Vec<i32> = scan.next_vec(n);
    let mut edge: Vec<Vec<usize>> = vec![Vec::new(); n];
    for _ in 0..n - 1 {
        let u: usize = scan.next();
        let v: usize = scan.next();
        let (u, v) = (u - 1, v - 1);
        edge[u].push(v);
        edge[v].push(u);
    }
    let mut dp: Vec<i32> = vec![i32::max_value(); n];
    let mut ans: Vec<usize> = vec![0; n];
    dfs(0, 0, &mut dp, &mut ans, &a, &edge);
    for x in ans {
        println!("{}", x);
    }
}