Extract a binary cotree from the width-zero contraction sequence and order its leaves along recursively chosen heavy paths. A light subtree attached at a join node contributes a constant $true$ row, while one attached at a union node contributes a constant $false$ row. Their placement makes these constant rows coalesce with canonical boundary values. Thus unequal-rank nodes are free, equal-rank nodes cost at most four changes, and the number of costly levels is the Strahler rank, at most $ceil(log₂ n)$.

The cotree ordering follows the logarithmic cograph-contiguity construction of Crespelle and Gambette; the framed-row invariant and its Lean proof are given here directly for Welzl crossing number.