Galton-Watson processes, simple varieties of trees and Khinchin families

Abstract In this note, we introduce a unified analytic framework that connects Bienaymé–Galton–Watson processes, simple varieties of trees and Khinchin families. Using Lagrange’s inversion formula, we derive new coefficient-based expressions for extinction probabilities and reinterpret them as boundary phenomena tied to the domain of the inverse of the solution to Lagrange’s equation. This perspective reveals an additional link between combinatorial and probabilistic models, simplifying classical arguments and yielding new results. It also provides a practical coefficient-based procedure: once the relevant power-series coefficients have been computed (e.g. via Lagrange inversion), the associated Galton–Watson probabilities can then be computed directly from these coefficients throughout a one-parameter family of distributions.