Overview

The Galton-Watson branching process is a second worked instance of a Markov chain — population size is itself a Markov chain on $\{0,1,2,\ldots\}$ — chosen because it introduces a genuinely new tool, the probability generating function, needed precisely because the state space is infinite and the matrix methods of Discrete-Time Markov Chains do not directly apply. The topic is deliberately short: two lessons are enough to state Galton's 1873 extinction question, answer it completely via a generating-function fixed-point argument, and classify the three qualitatively different regimes the answer produces.