The $O(\log n)$ Chain Property

Claim: Any path from the root to a node crosses at most $O(\log n)$ light edges. Proof: When you go down a light edge from $v$ to $u$, you enter a subtree with size[u] <= size[v] / 2 because $u$ was not the heaviest child. Each light edge halves the subtree size.

Since subtree size starts at $n$ and ends at $1$, you cross at most $\log_2 n$ light edges. Each light edge moves you to a new chain. So you visit at most $O(\log n)$ chains on any root-to-node path.

Space complexity is $O(n)$ for the data structures used.