Example: Finding a Centroid

Tree: $1-2$, $1-3$, $3-4$, $3-5$. Five nodes total. Start at node $1$. Its subtrees have sizes $1$ (node $2$) and $3$ (rooted at $3$). Node $3$'s subtree exceeds $5/2 = 2.5$, so move to node $3$. From node $3$, its children $4$ and $5$ each have size $1$, and the parent direction has size $2$ (nodes $1$ and $2$).

No component exceeds $2.5$. Node $3$ is the centroid. You walked from root to centroid in one step. The algorithm terminated quickly because the tree is small and well-balanced.