A local limit theorem for the number of nodes, the height, and the number of final leaves in a critical branching process tree

Let Z i be the number of particles in the i th generation of a non‐degenerate critical Bienaymé‐Galton‐Watson process with offspring distribution \documentclass{article}\pagestyle{empty}\begin{document}$ p_r = P \{\hbox{a given individual has {\it r} children}\},\kern2em r\geq 0. $\end{document} Let ν = Σ infinity 0 Z j be the total progeny and let ζ = inf{ r : Z r = 0} be the extinction time. Equivalently, ν and ζ are the total number of nodes and (1 + the height), respectively, of the family tree of the branching process. Assume that E { Z 1 } = Σ p r r = 1 and E { Z 1 3 + δ } = Σ p r r 3 + δ < infinity for some δ ϵ (0, 1). We find an asymptotic formula with remainder term for k 4 P {ζ = k + 1, Z k = ℓ ν = n } when k → infinity, which is uniform over n and ℓ. This is used to confirm a conjecture by Wilf that the number of leaves in the last generation of a randomly chosen rooted tree converges in distribution. More precisely, in the terminology introduced above, there exists a probability distribution { q 1 } such that for n → infinity \documentclass{article}\pagestyle{empty}\begin{document}$ P\{Z_{\zeta-1} = l | \nu=n\} = q_l + O \left({{\log^3 n } \over {n^{1/2}}}\right), $\end{document} uniformly over ℓ ≥1. The limiting distribution is identified by means of a functional equation for the generating function Σ infinity 1 q s ℓ . Numerically, q 1 ≅ 0.0602, q 2 ≅ 0.248, q 3 ≅ 0.094, and q 4 ≅ 0.035. Our method can also be used to find lim k → infinity k 4 P {ζ = k + 1, Z k = ℓ ν = n } when only E { Z 1 2 + δ } < infinity for some 0 ≤δ≤1, but we do not treat this case here; it goes without saying that the fewer moment assumptions one makes, the poorer the estimates become. © 1996 John Wiley & Sons, Inc.