Trees Generated by a Simple Branching Process

In this paper we consider the boundary behaviour of a tree generated by a simple branching process. The size (in the sense of Hausdorff measure) of the boundary is intimately related to the local asymptotic behaviour of the branching process and to the limit distribution of the maximum of a certain embedded Markov chain. Geometrically we can think of the tree as being a random set constructed in the following manner. Initially we are given a sphere and at the n-th stage each sphere is replaced by (or gives birth to) a random number of disjoint subspheres. The numbers are chosen independently of each other and of earlier choices but according to a fixed distribution. The limiting set (called a branching set) is topologically equivalent to the boundary of the tree. Branching sets arise in more than one context: we relate our theory to certain mathematical models for turbulence.