[树] hdu6035 Colorful Tree (2017 Multi-University Training Contest - Team 1 C)

@(ACM题目)[树]

Description

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.

Input

The input contains multiple test cases.
For each test case, the first line contains one positive integers n, indicating the number of node. (2≤n≤200000)
Next line contains n integers where the i-th integer represents ci, the color of node i. (1≤ci≤n)
Each of the next n−1 lines contains two positive integers x,y (1≤x,y≤n,x≠y), meaning an edge between node x and node y.
It is guaranteed that these edges form a tree.

Output

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

题目分析

本题给定一棵n个点的树，树上每个结点都有一个颜色，对树上的每条路径，统计其包含颜色数量。求所有路径上的这个数量的和。
为了后面叙述方便，我们将“以i结点为根节点的子树”称为$sub-tree_i$
记树上路径总数为m=n(n−1)2