Start at the root. If it is a centroid, you are done. If not, exactly one subtree has more than nodes. Why?
Because the root itself does not qualify, so one side must be heavy. Move into that heavy subtree and repeat. You will eventually reach a node where no subtree is heavy. That is a centroid. The walk cannot go on forever. This works because the heavy subtree keeps getting smaller as you walk deeper. Each step moves you closer to the balance point.