On Sibuya trees and forests

TL;DR

This work shows that the Sibuya distribution naturally governs the offspring laws in BGW processes and in both simply generated and increasing critical trees and forests. It develops a comprehensive generating-function framework, leveraging Gauss hypergeometric pgfs and Lagrange inversion, and reveals how generalized Stirling numbers drive the resulting occupancy distributions. The paper establishes explicit combinatorial structures (via $\mathcal{C}_{n,k}$ and $\mathcal{S}_{n,k}$), analyzes thermodynamic limits, and uncovers deep connections to Ewens–Pitman sampling and PD models in increasing trees. These results provide a unified view of the recursive formation and asymptotics of large forests, with implications for combinatorial probability and phylogenetic modeling.

Abstract

We show that the Sibuya distribution and its non-critical relatives are relevant in the context of the recursive generation of both simply generated and increasing critical trees' and forests' progenies. A special class of generalized Stirling numbers are at the heart of the analysis of the induced occupancy distributions. Asymptotic aspects of large forests are addressed.