Colorful Tree

Time Limit: 6000/3000 MS (Java/Others)

Memory Limit: 131072/131072 K (Java/Others)


There is a tree with $n$ nodes, each of which has a type of color represented by an integer, where the color of node $i$ is $c_i$.

The path between each two different nodes is unique, of which we define the value as the number of different colors appearing in it.

Calculate the sum of values of all paths on the tree that has $\frac{n(n-1)}{2}$ paths in total.


The input contains multiple test cases.

For each test case, the first line contains one positive integers $n$, indicating the number of node. $(2 \leq n \leq 200000)$

Next line contains $n$ integers where the $i$-th integer represents $c_i$, the color of node $i$. $(1 \leq c_i \leq n)$

Each of the next $n - 1$ lines contains two positive integers $x, y$ $(1 \leq x, y \leq n, x \neq y)$, meaning an edge between node $x$ and node $y$.

It is guaranteed that these edges form a tree.


For each test case, output "Case #$x$: $y$" in one line (without quotes), where $x$ indicates the case number starting from $1$ and $y$ denotes the answer of corresponding case.

Sample Input

3 1 2 1 1 2 2 3 6 1 2 1 3 2 1 1 2 1 3 2 4 2 5 3 6

Sample Output

Case #1: 6 Case #2: 29




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