Heavy and Light Edges

For each node $v$ with children, pick the child $u$ with the largest size[u]. The edge from $v$ to $u$ is heavy. All other edges from $v$ are light. If there is a tie for largest child, pick any one arbitrarily. Leaf nodes have no children, so no edges from them need classification.

This simple rule guarantees the $O(\log n)$ chain property. Every node has at most one heavy edge going down to a child. A node can have multiple light edges down, but only one heavy edge.

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