An examination of the difficulties that statistical theory and, in particular, estimation theory can encounter within the area of dependent data. This is achieved through the study of the theory of branching processes starting with the demographic question: what is the probability that a family name becomes extinct? Contains observations on the generation sizes of the Bienayme-Galton-Watson (BGW) process. Various parameters are estimated and branching process theory is contrasted to a Bayesian approach. Illustrations of branching process theory applications are shown for particular problems.