November 2011
Problem 4
Bear Trees

Gary is a bear. He lives in a system of caves, consisting of N caverns, numbered from 0 through N – 1. These caverns are connected by bidirectional tunnels, such that there is exactly one path between each pair of tunnels. (You might also know this kind of structure as a "tree", so you’ll know that there are necessarily N – 1 tunnels.)

Gary would like to explore this system of caves, using the following method:

  1. Put cavern 0 (his home) on a "to explore" list.
  2. Choose one cavern C from the list.
  3. Remove C from the list.
  4. Explore C: Add all caverns adjacent to C that have never been on the list.
  5. Repeat steps 2 to 3 while the list contains at least one cavern.

There are many ways to explore a system of caves. However, bears are forgetful. You would like to find a way to explore the cave such that the maximum length of the list is minimized. For example, given the following tree:

Here is one possible way to explore the tree, where the maximum length of the list is 4:

However, exploring in a different order, Gary can make it such that he never has to remember more than 3 elements; indeed, it is easy to see that 3 is optimal. Gary has retained you to find this minimum list length, given a system of caves.

The input file DATA5.txt will contain 5 test cases. Each test case will begin with one line, containing the number of caverns 1 <= N <= 1000. N – 1 lines will follow, each consisting of two distinct space-separated integers x and y, denoting a tunnel between caverns x and y. Of course, no tunnel will be described more than once, and 0 <= x, y < N.

The output file OUT5.txt will contain 5 lines of output, the minimum list length for each cave system.

Sample Input (2 cases shown):
0 1
0 2
0 3
2 4
2 5
3 6
0 1
1 2
2 3
Sample Output (2 cases shown):