On the complexity of computing Strahler numbers

Abstract

It is shown that the problem of computing the Strahler number of a binary tree given as a term is complete for the circuit complexity class uniform ${\mathsf{NC}}^1$. For several variants, where the binary tree is given by a pointer structure or in a succinct form by a directed acyclic graph or a tree straight-line program, the complexity of computing the Strahler number is determined as well. We show that the problem of deciding whether a given context-free grammar in Chomsky normal form produces a derivation tree with a Strahler number of at least $k$ is $\mathsf{P}$-complete. If the derivation tree is restricted to be acyclic, the problem becomes $\mathsf{PSPACE}$-complete.

Extended abstract appeared at STACS 2026