Given a tree, I need to find all the edges that are shared by all diameters of that tree. In other words, let D be the length of the longest path in the tree and P be the set of all paths in the tree with length = D. I need to find the intersection of the edges shared by all paths in P. How can I do that? Please do not just mention the algorithm, I would appreciate if you can share the theoretical justification of why it works too (links to lectures or any resources are also very appreciated).

Thank you very much.