Finding Center of The Tree

Question

I have a question which is part of my program.

For a tree T=(V,E) we need to find the node v in the tree that minimize the length of the longest path from v to any o

Answer 1

Consider a tree with two nodes? Which is the center? Either one will suffice, ergo a tree can have more than one center.

Now, think about what it means to be the center. If all of the branches are the same height, the center is the root (all paths go through the root). If the branches are of different heights then the center must be either the root or in the branch with the greatest height otherwise the maximum path is greater than the height of the tallest branch and the root would be a better choice. Now, how far down the tallest branch do we need to look? Half the difference in height between the tallest branch (from the root) and the next tallest branch (if the difference is at most 1 the root will suffice). Why, because as we go down the tallest branch by one level we are lengthening the path to the deepest node of the next tallest branch by one and reducing the distance to the deepest node in the current branch by one. Eventually, they will be equal when you've traversed half the difference in the depth. Now as we go down the tallest branch, we simply need to choose at each level the tallest sub-branch. If more than one has the same height, we simply choose one arbitrarily.

Essentially, what you are doing is finding the longest path in the graph, which is the distance between the tallest branch of the tree and the next tallest branch, then finding the middle node on that branch. Because there may be multiple paths of the same length as the longest path and the length of the longest path may be even, you have multiple possible centers.