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 Python (3.8.2) (judge verdict: Runtime Error)

n = int(input())
c = [int(ci) - 1 for ci in input().split()]
v = [[] for _ in range(n)]
for _ in range(n - 1):
    ai, bi = map(lambda x: int(x) - 1, input().split())
    v[ai].append(bi)
    v[bi].append(ai)

par = [-1] * n
ch = [[] for _ in range(n)]
def sep_parnch(x=0, parent=-1):
    par[x] = parent
    for i in v[x]:
        if i == parent:
            continue
        ch[x].append(i)
        sep_parnch(i, x)
    return

sep_parnch()

tsize = [0] * n
def fill_tsize(root=0):
    tsize[root] = sum(fill_tsize(x) for x in ch[root]) + 1
    return tsize[root]

fill_tsize()

seg_size = [[0] * len(ch[x]) for x in range(n)]
def fill_seg_size(x=0, clist=[(None, -1)]*n):
    al, aind = clist[c[x]]
    if al is not None:
        al[aind] += tsize[x]
    for i, y in enumerate(ch[x]):
        clist[c[x]] = seg_size[x], i
        fill_seg_size(y, clist)
    clist[c[x]] = al, aind
    return

head_seg = [([0], 0) for _ in range(n)]
fill_seg_size(0, head_seg)

def comb2(a):
    return a * (a - 1) // 2 + a

vcat = [[] for _ in range(n)]
for i in range(n):
    vcat[c[i]].append(i)

tot = comb2(n)
for col, vs in enumerate(vcat):
    ans = sum(comb2(tsize[y] - seg_size[x][i]) for x in vs for i, y in enumerate(ch[x])) + comb2(n - head_seg[col][0][0])
    print(tot - ans)