Extensions

You can extend LCA to answer more complex queries: kth ancestor (jump to the ancestor $k$ steps above $u$), path median (find the middle node on a path), and heavy-light decomposition for path updates. With binary lifting, the kth ancestor is a binary representation query: jump $u$ upward by the powers of two in $k$'s binary form.

These extensions turn LCA from a simple query into a toolkit for tree algorithms.