On family trees and subtrees of simple branching processes (Q808106)

The family tree in a simple branching process initiated by a single ancestor is said to contain an N-ary subtree if a subtree can be found in which every individual has at least N offsprings. The second author [Am. Math. Monthly 98, 728-731 (1991)] previously studied the comparatively tractable case \(N=2\), showing how to obtain \(\tau\) (2), the probability of an infinite binary subtree. This paper studies the analogous \(\tau\) (N) for general N, obtaining a number of partial results and specialising to standard forms of offspring distributions such as Poisson for more detailed estimation, using numerical calculations for illustration.