2023/07/12 by Khanfir, Robin · 1 citation
#60E07 #60F05. Secondary: 05C05 #60J80 #FOS: Mathematics #Primary: 60C05 #Probability (math.PR)
paper · doi:10.48550/arxiv.2307.05983
The Horton-Strahler number, also known as the register function, provides a tool for quantifying the branching complexity of a rooted tree. We consider the Horton-Strahler number of critical Galton-Watson trees conditioned to have size n and whose offspring distribution is in the domain of attraction of an α-stable law with α∈ [1, 2]. We give tail estimates and when α≠ 1, we prove that it grows as \frac1αlogα/(α-1) n in probability. This extends the result in Brandenberger, Devroye & Reddad [6] dealing with the finite variance case for which α=2. We also characterize the cases where α=1, namely the spectrally positive Cauchy regime, which exhibits more complex behaviors. Our proofs are new and probabilistic; they relate the Horton-Strahler number with other shape parameters such as the height or largest degree.