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Let Z, be the number of particles in the ith generation of a non-degenerate critical BienaymC-Galton-Watson process with offspring distribution p , = P{a given individual has r children} , r 2 0 Let u = C : Z j be the total progeny and let s=inf{r:Z,=O} be the extinction time. Equivalently, u and C are the total number of nodes and (1 + the height), respectively, of the family tree of the branching process. Assume that E { Z , } = C p,r = 1 and E{Z:'"} = C prr3+' < m for some 6 E (0,l). We find an asymptotic formula with remainder term for k4P{ 5 = k + 1, Z , = 8, v = n} when k+ m, which is uniform over n and f. 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 (4,) such that for n-+m P{ z, -, = e I Ir = n } = qr + 0 (9) , uniformly over f r l . The limiting distribution is identified by means of a functional equation for the generating function ,Yy qp'. Numerically, q1 = 0.602, q2 = 0.248, q3 = * Research supported by the NSF through a grant to Cornell University. Random Structures and Algorithms, Vol. 8, No. 4 (1996)

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