The Horton-Strahler number of butterfly trees

The Horton-Strahler number (HS) is a measure of branching complexity of rooted trees, introduced in hydrology and later studied in parallel computing under the name register function. While its order of growth is well understood for classical random trees, fluctuation behavior has largely resisted analysis. In this work we investigate the HS in the setting of butterfly trees -- binary trees constructed from butterfly permutations, a rich class of separable permutations with origins in numerical linear algebra and parallel architectures. For the subclass of simple butterfly trees, we exploit their recursive gluing structure to model the HS as an additive functional of a finite-state Markov process. This framework yields sharp distributional results, including a law of large numbers and a Central Limit Theorem with explicit variance growth, providing what appears to be the first genuine Gaussian limit law for the HS in a nontrivial random tree model. Extending to biased constructions, we further establish functional limit theorems via analytic and probabilistic tools. For general butterfly trees, while exact analysis remains open, empirical sampling shows that the HS distribution is confined to a narrower support than in classical models, and appears to concentrate tightly near the upper bound $\lfloor \log_4 N\rfloor$.