Galton-Watson Process

A Galton-Watson process is a discrete-time branching process in which every individual independently produces a number of offspring having the same probability distribution. If is the population in generation and the offspring generating function is , then the generating function of starting from one individual is the -fold iterate . The extinction probability is the smallest solution of

If the mean number of offspring is at most 1, extinction occurs with probability 1 except in the degenerate case almost surely.