A. RELATIONSHIPS BETWEEN RANDOM TREES AND BRANCHING PROCESSES

Let z be a non-negative, integer-valued random variable and let z i, 1 i < oo, be independent random variables, all having the same distribution as z. At t = 0 one particle is alive, and at t = 1 it gives birth to a random number z 1 of offspring. At t = 2 each of the z 1 first-generation particles gives birth to a random number (z 2, z 3,th.z +1) of offspring, etc. If N(k) is the number of particles in the k generation, 1 then N(k), k = 0, 1, 2,..., is called a branching process. Letting z denote the expected value of z, it can be shown I that-k N(k) = z. (1