| In a certain business sector there are currently N small companies, each |
| having just a single employee. These employees are numbered 1 through N. |
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| 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. |
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| 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. |
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| 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. |
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| ### Input |
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| The first line contains the number of test cases T. |
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| Each test case starts with a blank line. The next line contains two integers, |
| N and D. |
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| 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. |
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| The mergers must be performed in the order in which they are given. |
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| ### Constraints |
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| 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. |
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| ### Output |
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| 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. |
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| ### Examples |
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| 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. |
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