Abstract
Classical branching-process theory, developed by Galton and Watson in the nineteenth century and later refined by Fisher and Haldane, provides the formal framework for quantifying the fate of new mutants, new viral and bacterial pathogens, new colonization of invasive species, etc. It is a powerful tool to quantify and predict the effect of differential reproductive success on the speciation potential of evolutionary lineages. Here, I revisit the conceptual framework of the branching process, detail its mathematical development over time, tie up a few historical loose strings, illustrate the calculation of the exact extinction probability for the Poisson-distributed reproductive success with the Lambert function (which is often missing in the ecological and evolutionary literature), and highlight the potential applications of the branching process in modern ecology and evolutionary biology, especially in deriving the extinction probability and extinction time. I also highlight a few misconceptions about human demography in the US that can be readily dismissed by applying probability tools such as branching processes.