Start at any node. If it is a centroid, you are done. If not, one of its subtrees has more than nodes. Move into that subtree and repeat. You cannot loop forever because each step decreases the maximum subtree size. Eventually you reach a node where no subtree exceeds . That is your centroid.
The parent direction also forms a component, but it has fewer than nodes (otherwise you would not have moved here). This proof also gives you an algorithm: walk toward the heavy subtree until you find balance. It is constructive and efficient.