Walkthrough

Tree: $1-2$, $2-3$, $2-4$, $4-5$. Five nodes. Root at $1$. After DFS, subtree sizes: size[1]=5, size[2]=4, size[3]=1, size[4]=2, size[5]=1. Node $1$ has child $2$ with size[2]=4 > 5/2=2.5. Move to node $2$. From $2$, children are $1$ (size $1$), $3$ (size $1$), and $4$ (size $2$).

None exceed $2.5$. The parent direction would have size $1$, also under the threshold. Node $2$ is the centroid. Output: $2$. The algorithm terminated in two steps, demonstrating the efficiency of walking toward heavy subtrees.