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 Pascal (FPC 3.0.4) (judge verdict: Accepted)

var
	N,k:int64;
	i,u,v,es:Longint;
	VV,nxt,beg:Array[1..400000]of Longint;
	ans:Array[1..200000]of int64;
	cnt,c:Array[1..200000]of Longint;
function dfs(u,p:Longint):Longint;
var
	ret,sto,pre,id,t:Longint;
	k:int64;
begin
	ret:=1;
	sto:=cnt[c[u]];
	pre:=sto;
	id:=beg[u];
	while id<>0 do begin
		if VV[id]<>p then begin
			t:=dfs(VV[id],u);
			inc(ret,t);
			k:=t-(cnt[c[u]]-sto);
			sto:=cnt[c[u]];
			dec(ans[c[u]],k*(k+1)div 2);
		end;
		id:=nxt[id];
	end;
	cnt[c[u]]:=pre+ret;
	dfs:=ret;
end;
begin
	read(N);
	for i:=1 to N do read(c[i]);
	for i:=2 to N do begin
		read(u,v);
		inc(es);
		VV[es]:=v;
		nxt[es]:=beg[u];
		beg[u]:=es;
		inc(es);
		VV[es]:=u;
		nxt[es]:=beg[v];
		beg[v]:=es;
	end;
	dfs(1,0);
	for i:=1 to N do begin
		inc(ans[i],N*(N+1)div 2);
		k:=N-cnt[i];
		dec(ans[i],k*(k+1)div 2);
		writeln(ans[i]);
	end;
end.