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 C++ (GCC 9.2.1) (judge verdict: Runtime Error)

#include<bits/stdc++.h>
using namespace std;
using UL = unsigned long;
using ULL = unsigned long long;
using LL = long long;
#define rep(i, n) for(int i = 0; i < (n); i++)

struct Problem {
    UL N;
    vector<UL> C;
    vector<vector<UL>> E;

    vector<ULL> ans;
    vector<UL> F;

    void Concat(map<UL, UL>& d, const map<UL, UL>& s) {
        for (auto i : s) d[i.first] += i.second;
    }

    pair<UL, map<UL, UL>> DFS(UL p, map<UL, UL>& back) {
        F[p] = 1;
        map<UL, UL> tmp;
        UL Vcount = 0;
        for (UL e : E[p]) {
            if (F[e]) continue;
            auto buf = DFS(e, tmp);
            ULL backN = buf.first - buf.second[C[p]];
            ULL forwdN = Vcount - tmp[C[p]];
            ans[C[p]] += backN * forwdN;
            Concat(tmp, buf.second);
            Vcount += buf.first;
        }
        Vcount++;
        ULL backN = N - Vcount + 1;
        ULL forwdN = Vcount - tmp[C[p]];
        ans[C[p]] += backN * forwdN;
        tmp[C[p]] = Vcount;
        /*
        cout << "p=" << p << ", Vcount=" << Vcount;
        cout << ", ans={"; rep(i, N) cout << ans[i] << ","; cout << "}";
        cout << ", tmp={"; rep(i, N) cout << tmp[i] << ","; cout << "}";
        cout << ", back={"; rep(i, N) cout << back[i] << ","; cout << "}" << endl;
        */
        return { Vcount, move(tmp) };
    }

    void Solve() {
        scanf("%d", &N);
        C.resize(N); rep(i, N) { scanf("%d", &C[i]); C[i]--; }
        E.resize(N);
        rep(i, N - 1) {
            UL U, V; scanf("%d%d", &U, &V); U--; V--;
            E[U].push_back(V);
            E[V].push_back(U);
        }
        ans.resize(N);
        F.resize(N);
        map<UL, UL> r;
        DFS(0, r);
        rep(i, N) cout << ans[i] << endl;
    }
};
int main() {
    unique_ptr<Problem> p(new Problem());
    p->Solve();
    return 0;
}