Google Code Jam 2014 Round 1 A：Problem B. Full Binary Tree

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: X_i Y_i, indicating that G contains an undirected edge between X_i and Y_i.

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

1 ≤ T ≤ 100. 1 ≤ X_i, Y_i ≤ N Each test case will form a valid connected tree.

Small dataset

2 ≤ N ≤ 15.

Large dataset

2 ≤ N ≤ 1000.

Sample

Input	Output
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).