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

Abstract

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 ) 0.094, and q4 = 0.035. Our method can also be used to find Iim,-= k4P{ 6 = k + 1, Z , = f, Y = n} when only E { Z : ' 6 } < m for some 0 5 6 5 1, but we do not treat this case here; it goes without saying that the fewer moment assumptions one makes, the poorer the error estimates become. 0 19% John Wiley & Sons, Inc.

Pattern Avoiding Permutations with a Unique Longest Increasing Subsequence

“…A very general theorem, that contains this result, is Theorem 2 in [8]. An earlier, and more specific, reference is [7].…”

On the number of vertices with a given degree in a Galton-Watson tree

“…Now suppose that f (z) is supercritical and let q be the extinction probability: f (q) = q. Then we can take q 1 = q, q 2 = 1 and (19) and (20) are proved in this case. When f (z) is critical, or subcritical with ρ = 1, we have a = f (a) = b k = g k (b k ) = 1 and there is nothing to be proved.…”

“…For other types of conditional limit theorem considered on the event {Z = n}, see, e.g. [18] and [19].…”

On a random search tree: asymptotic enumeration of vertices by distance from leaves

“…See [8] for a comprehensive treatment of these results. The tree height and width are parameters of a global nature; see, for example, [3], [11]- [13], [17], and [18]. Profiles of random trees have been studied in [6] and [16].…”

“…See Flajolet and Sedgewick [7] for a comprehensive treatment of these results. The tree height and width are parameters of global nature, see Kolchin [9], Devroye [3], Mahmoud and Pittel [10], Pittel [14], Kesten and Pittel [8], Pittel [15], for instance. Profiles of random trees have been studied in [4] and [13].…”