| In a certain business sector there are currently N small companies, each | |
| having just a single employee. These employees are numbered 1 through N. | |
| The business sector is about to be transformed into a monopoly. This will | |
| happen through a series of mergers, until there is only one company. A single | |
| merger involves two companies. In a merger, the president of one company | |
| becomes the direct report of the president of the other company, preserving | |
| the rest of the hierarchies of both companies. | |
| You will be given the descriptions of all mergers. Depending on how they are | |
| performed (which of the two presidents involved becomes the president of the | |
| new company), the hierarchy can of the final company can take different | |
| shapes. We want the hierarchy of the final company to be as shallow as | |
| possible. The task is to find the smallest possible number of levels in the | |
| final hierarchy. | |
| There is also a limit D on the number of direct reports any employee can have. | |
| Because of this limit, there may be only one way to accomplish a certain | |
| merger, or it might even be impossible. However, there will always be some way | |
| to accomplish all the mergers. | |
| ### Input | |
| The first line contains the number of test cases T. | |
| Each test case starts with a blank line. The next line contains two integers, | |
| N and D. | |
| Each of the following N-1 lines describes a single merger, with two integers | |
| between 1 and N. These are the employees whose companies are merging. The two | |
| employees will never already be part of the same company. | |
| The mergers must be performed in the order in which they are given. | |
| ### Constraints | |
| 5 ≤ T ≤ 20 | |
| 2 ≤ N ≤ 30,000 | |
| 1 ≤ D ≤ 5,000 | |
| The input test cases will be such that it is possible to accomplish all | |
| mergers. | |
| ### Output | |
| For each of the test cases numbered in order from 1 to T, output "Case #i: " | |
| followed by a single integer, the smallest number of levels in the final | |
| hierarchy. | |
| ### Examples | |
| In the first example, we have N=3 and D=2. The first merger happens between | |
| the companies of employees 1 and 2. In the resulting company we can have | |
| employee 1 as the president with 2 as his report, or vice versa. Next this | |
| company merges with the company of employee 3. If we have employee 3 become | |
| the president, the hierarchy will be a chain 3-1-2 or 3-2-1. If 1 or 2 become | |
| the president, that president will have the other two employees as direct | |
| reports. This last hierarchy has two levels. | |