Problem
A tree is a connected graph with no cycles.
A rooted tree is a tree in which one special vertex is called the root. If there is an edge between X and Y in a rooted tree, we say that Y is a child of X if X is closer to the root than Y (in other words, the shortest path from the root to X is shorter than the shortest path from the root to Y).
A full binary tree is a rooted tree where every node has either exactly 2 children or 0 children.
You are given a tree G with N nodes (numbered from 1 to N). You are allowed to delete some of the nodes. When a node is deleted, the edges connected to the deleted node are also deleted. Your task is to delete as few nodes as possible so that the remaining nodes form a full binary tree for some choice of the root from the remaining nodes.
Input
The first line of the input gives the number of test cases, T. T test cases follow. The first line of each test case contains a single integer N, the number of nodes in the tree. The following N-1 lines each one will contain two space-separated integers: Xi Yi, indicating that G contains an undirected edge between Xi and Yi.
Output
For each test case, output one line containing "Case #x: y", where x is the test case number (starting from 1) and y is the minimum number of nodes to delete from G to make a full binary tree.
Limits
Memory limit: 1 GB.
1 ≤ T ≤ 100.
1 ≤ Xi, Yi ≤ N
Each test case will form a valid connected tree.
Small dataset (9 Points)
Time limit: 60 5 seconds.
2 ≤ N ≤ 15.
Large dataset (21 Points)
Time limit: 120 10 seconds.
2 ≤ N ≤ 1000.
Sample
3 3 2 1 1 3 7 4 5 4 2 1 2 3 1 6 4 3 7 4 1 2 2 3 3 4
Case #1: 0 Case #2: 2 Case #3: 1
In the first case, G is already a full binary tree (if we consider node 1 as the root), so we don't need to do anything.
In the second case, we may delete nodes 3 and 7; then 2 can be the root of a full binary tree.
In the third case, we may delete node 1; then 3 will become the root of a full binary tree (we could also have deleted node 4; then we could have made 2 the root).