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 ith generation of a non-degenerate critical Bienaymé-Galton-Watson process with offspring distribution $ p_r = P \{\hbox{a given individual has {\it r} children}\},\kern2em r\geq 0. $ Let ν = Σ^infinity₀ 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{Z₁} = Σ p_rr = 1 and E{Z₁^{3 + δ}} = Σ p_rr^{3 + δ} < infinity for some δ ϵ (0, 1). We find an asymptotic formula with remainder term for k⁴P{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 {q₁} such that for n → infinity $ P\{Z_{\zeta-1} = l | \nu=n\} = q_l + O \left({{\log^3 n } \over {n^{1/2}}}\right), $ uniformly over ℓ ≥1. The limiting distribution is identified by means of a functional equation for the generating function Σ^infinity₁ q s^ℓ. Numerically, q₁ ≅ 0.0602, q₂ ≅ 0.248, q₃ ≅ 0.094, and q₄ ≅ 0.035. Our method can also be used to find lim _{k→ infinity} k⁴P{ζ = k + 1, Z_k = ℓ ν = n} when only E{Z₁^{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.