the offspring generating function
The offspring generating function is the algebraic engine that turns the Galton-Watson process from a verbal description into a computation. The deep reason a branching process is so tractable is that composing generations corresponds to composing one fixed function. If you can write down the generating function of the offspring law, you can in principle read off extinction probabilities, growth rates, and the whole distribution of the population at every generation by iterating it.
For an offspring law (p_k){k>=0}, define the probability generating function f(s) = E[s^X] = sum_(k>=0) p_k s^k for s in [0, 1]. It is increasing, convex, with f(1) = 1 and f'(1) = m, the mean offspring number (when finite). The central identity is the composition rule: if f_n(s) = E[s^(Z_n)] is the generating function of the n-th generation (starting from one ancestor), then f_n is the n-fold composition of f with itself, f_n = f o f o ... o f (n times). This follows because the population in generation n+1 is a sum of Z_n independent subpopulations each launched by an individual of generation n; summing independent random variables multiplies generating functions, and randomizing the number of summands composes them. Hence the entire law of Z_n is encoded by iterating one map of the interval [0, 1].
Why it matters: this single fact reduces a multi-generation stochastic problem to the dynamics of a one-dimensional map, the cleanest example of a self-similar probabilistic structure. The extinction probability is the smallest fixed point of f; the mean growth is f'(1)^n by the chain rule; and the Kesten-Stigum and Yaglom limit theorems are ultimately statements about the iterates f_n near their fixed point. One caveat: the generating function determines the law on the nonnegative integers uniquely, but its derivatives at 1 give moments only if those moments are finite — f'(1) = infinity signals an infinite-mean (and typically supercritical-in-spirit) process where the standard m-classification must be handled with care.
For a geometric-with-failure offspring law p_k = (1-p) p^k, the generating function is f(s) = (1-p)/(1 - p s), a Mobius (fractional-linear) map. Mobius maps compose neatly, so f_n is again fractional-linear, and one can write down the exact distribution of Z_n in closed form — a rare luxury that makes this the textbook example.
Generations compose as functions: f_n = f composed with itself n times.
The pgf determines the integer law uniquely, but f'(1) equals the mean ONLY when the mean is finite; for heavy-tailed offspring f'(1) = infinity and the usual mean-based classification needs care.