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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Runtime.CompilerServices;
using System.Text;
using System.Diagnostics;
using System.Numerics;
using static System.Console;
using static System.Convert;
using static System.Math;
using static Template;
using Pi = Pair<int, int>;

class Solver
{
    int N;
    long[] res;
    List<int>[] g;
    int[][] cc;
    int[] ch,col;
    List<int>[] c;
    public void Solve(Scanner sc)
    {
        N = sc.Int;
        g = Create(N, () => new List<int>());
        col = sc.ArrInt.Select(v => --v).ToArray();
        for (int i = 0; i < N-1; i++)
        {
            int a, b;
            sc.Make(out a, out b);a--;b--;
            g[a].Add(b);g[b].Add(a);
        }
        ch = new int[N];res = new long[N]; 
        c = Create(N, () => new List<int>() { -1 });
        cc = new int[N][];
        child(0, -1);
        rm(0, -1);
        for (int i = 0; i < N; i++)
        {
            var ct = N;
            while (c[i].Last() != -1)
            {
                var v = c[i].PopBack();
                ct -= ch[v];
            }
            res[i] += (long)ct * (ct - 1) / 2 + ct;
        }
        Console.WriteLine(string.Join("\n", res.Select(v=>(long)N*(N-1)/2+N-v)));
    }
    void child(int i,int p)
    {
        cc[i] = new int[g[i].Count];
        for (int j = 0; j < g[i].Count; j++)
        {
            int e = g[i][j];
            if (e == p) continue;
            child(e, i);
            cc[i][j] = ch[e];
            ch[i] += ch[e];
        }
        ch[i]++;
    }
    void rm(int i,int p)
    {
        int st = c[col[i]].Last();
        for(int j=0;j<g[i].Count;j++)
        {
            int e = g[i][j];
            if (e == p) continue;
            rm(e, i);
            while (c[col[i]].Last() != st)
            {
                var v = c[col[i]].PopBack();
                cc[i][j] -= ch[v];
            }
            res[col[i]] += (long)cc[i][j]*(cc[i][j]-1)/2+cc[i][j];
        }
        c[col[i]].Add(i);
    }
    void calc(int i,int p)
    {
        foreach (var e in g[i])
        {
            if (e == p) continue;
            calc(e, i);
        }
        long sum = 0;
        for(int j = 0; j < g[i].Count; j++)
        {
            sum += cc[i][j] * (cc[i][j] - 1) / 2;
        }
        res[col[i]] += sum;
    }
}

class LowestCommonAncestor
{
    private List<int>[] g;
    private int N, lb;
    private int[][] parent;
    private int[] depth;
    private bool finished;
    public LowestCommonAncestor(int N)
    { this.N = N; g = Create(N, () => new List<int>()); }
    public void AddEdge(int u, int v)
    {
        g[u].Add(v); g[v].Add(u);
    }
    public void Build(int root = 0)
    {
        for (lb = 31; lb >= 0; lb--)
            if ((1 & N >> lb) == 1)
                break;
        parent = Create(lb + 1, () => Create(N, () => -1));
        depth = new int[N];
        var st = new Stack<int>();
        st.Push(-1);
        st.Push(root);
        while (st.Any())
        {
            int i = st.Pop(), p = st.Pop();
            parent[0][i] = p;
            foreach (var e in g[i])
                if (e != p)
                {
                    depth[e] = depth[i] + 1;
                    st.Push(i); st.Push(e);
                }
        }
        for (var i = 1; i <= lb; i++)
            for (var j = 0; j < N; j++)
                if (parent[i - 1][j] != -1)
                {
                    parent[i][j] = parent[i - 1][parent[i - 1][j]];
                }
        finished = true;
    }
    public int LCA(int u, int v)
    {
        if (!finished) throw new Exception("You Need to Execute \"Build()\"");
        if (depth[u] > depth[v])
            swap(ref u, ref v);
        for (var i = lb; i >= 0; i--)
            if ((1 & (depth[v] - depth[u]) >> i) == 1)
                v = parent[i][v];
        if (u == v) return u;
        for (var i = lb; i >= 0; i--)
            if (parent[i][u] != parent[i][v])
            { u = parent[i][u]; v = parent[i][v]; }
        return parent[0][u];
    }
}
#region Template
public static class Template
{
    static void Main(string[] args)
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver().Solve(new Scanner());
        Console.Out.Flush();
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) == 1) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) == -1) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T a, ref T b) { var t = b; b = a; a = t; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(this IList<T> A, int i, int j) { var t = A[i]; A[i] = A[j]; A[j] = t; }
    public static T[] Shuffle<T>(this IList<T> A) { T[] rt = A.ToArray(); Random rnd = new Random(); for (int i = rt.Length - 1; i >= 1; i--) swap(ref rt[i], ref rt[rnd.Next(i + 1)]); return rt; }
    public static T[] Create<T>(int n, Func<T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(); return rt; }
    public static T[] Create<T>(int n, Func<int, T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(i); return rt; }
    public static int CompareTo<T>(this T[] A, T[] B, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; for (var i = 0; i < Min(A.Length, B.Length); i++) { int c = cmp(A[i], B[i]); if (c > 0) return 1; else if (c < 0) return -1; } if (A.Length == B.Length) return 0; if (A.Length > B.Length) return 1; else return -1; }
    public static int MaxElement<T>(this IList<T> A, Comparison<T> cmp = null) { cmp = cmp ?? Comparer<T>.Default.Compare; T max = A[0]; int rt = 0; for (int i = 1; i < A.Count; i++) if (cmp(max, A[i]) < 0) { max = A[i]; rt = i; } return rt; }
    public static T PopBack<T>(this List<T> A){ var v = A[A.Count - 1]; A.RemoveAt(A.Count - 1);return v; }
    public static void Fail<T>(T s) { Console.WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
}
public class Scanner
{
    public string Str => Console.ReadLine().Trim();
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public int[] ArrInt => Str.Split(' ').Select(int.Parse).ToArray();
    public long[] ArrLong => Str.Split(' ').Select(long.Parse).ToArray();
    public char[][] Grid(int n) => Create(n, () => Str.ToCharArray());
    public int[] ArrInt1D(int n) => Create(n, () => Int);
    public long[] ArrLong1D(int n) => Create(n, () => Long);
    public int[][] ArrInt2D(int n) => Create(n, () => ArrInt);
    public long[][] ArrLong2D(int n) => Create(n, () => ArrLong);
    private Queue<string> q = new Queue<string>();
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T Next<T>() { if (q.Count == 0) foreach (var item in Str.Split(' ')) q.Enqueue(item); return (T)Convert.ChangeType(q.Dequeue(), typeof(T)); }
    public void Make<T1>(out T1 v1) => v1 = Next<T1>();
    public void Make<T1, T2>(out T1 v1, out T2 v2) { v1 = Next<T1>(); v2 = Next<T2>(); }
    public void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3) { Make(out v1, out v2); v3 = Next<T3>(); }
    public void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4) { Make(out v1, out v2, out v3); v4 = Next<T4>(); }
    public void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5) { Make(out v1, out v2, out v3, out v4); v5 = Next<T5>(); }
    public void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6) { Make(out v1, out v2, out v3, out v4, out v5); v6 = Next<T6>(); }
    //public (T1, T2) Make<T1, T2>() { Make(out T1 v1, out T2 v2); return (v1, v2); }
    //public (T1, T2, T3) Make<T1, T2, T3>() { Make(out T1 v1, out T2 v2, out T3 v3); return (v1, v2, v3); }
    //public (T1, T2, T3, T4) Make<T1, T2, T3, T4>() { Make(out T1 v1, out T2 v2, out T3 v3, out T4 v4); return (v1, v2, v3, v4); }
}
public class Pair<T1, T2> : IComparable<Pair<T1, T2>>
{
    public T1 v1;
    public T2 v2;
    public Pair() { }
    public Pair(T1 v1, T2 v2)
    { this.v1 = v1; this.v2 = v2; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2> p)
    {
        var c = Comparer<T1>.Default.Compare(v1, p.v1);
        if (c == 0)
            c = Comparer<T2>.Default.Compare(v2, p.v2);
        return c;
    }
    public override string ToString() => $"{v1.ToString()} {v2.ToString()}";
    public void Deconstruct(out T1 a, out T2 b) { a = v1; b = v2; }
}

public class Pair<T1, T2, T3> : Pair<T1, T2>, IComparable<Pair<T1, T2, T3>>
{
    public T3 v3;
    public Pair() : base() { }
    public Pair(T1 v1, T2 v2, T3 v3) : base(v1, v2)
    { this.v3 = v3; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2, T3> p)
    {
        var c = base.CompareTo(p);
        if (c == 0)
            c = Comparer<T3>.Default.Compare(v3, p.v3);
        return c;
    }
    public override string ToString() => $"{base.ToString()} {v3.ToString()}";
    public void Deconstruct(out T1 a, out T2 b, out T3 c) { Deconstruct(out a, out b); c = v3; }
}
#endregion