Efficient all pair bottleneck computation for a tree
You can even get $O(|V|)$ time if the edge weights are sorted. See the following paper:

Erik D. Demaine, Gad Landau, and Oren Weimann, “On Cartesian Trees and Range Minimum Queries”, in Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP 2009), Lecture Notes in Computer Science, volume 5555, Rhodes, Greece, July 5–12, 2009, pages 341–353.