algorithms – Better version of heavy-light decomposition: always draw path to largest subtree

Can someone tell me if this works? It seems to be easier to implement and also creates slightly less paths (though still O(log N) ).

The problem: given any tree of size N, find a way to split it into some disjoint paths such that from any node, you can reach the root by passing through log N paths. Heavy light decomposition:

I was thinking that a small modification is that instead of choosing an edge only when a child’s subtree is of size at least half of this subtree, you just choose the largest subtree. Proof that this works is the same: when you switch paths, the size of the subtree must at least double.

Edit: just noticed that the link also mentions you can do this, so I’m wondering why heavy-light decomposition is used instead of this, because this seems easier to understand and also slightly more performant (because of less paths).