Core idea 2 - Walk to Heavy Subtree

Start at the root. If it is a centroid, you are done. If not, exactly one subtree has more than $n/2$ 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.