the extinction probability
The original question of branching theory is: does the lineage survive forever, or is it doomed to die out? The extinction probability q is the chance that the population eventually reaches 0 and stays there. The beautiful answer is that q is determined by a single equation — q is the smallest fixed point of the offspring generating function — and the geometry of that one convex function tells you everything.
Let f be the offspring generating function and m = f'(1) the mean. Extinction by generation n is the event {Z_n = 0}, whose probability is f_n(0) (plug s = 0 into the n-fold iterate). These probabilities increase to q = P(extinction ever) = lim_n f_n(0). Letting n -> infinity in the recursion f_(n+1)(0) = f(f_n(0)) shows q satisfies the fixed-point equation q = f(q). Because f is convex, increasing, with f(1) = 1, the equation s = f(s) on [0, 1] has either the single root s = 1 (when m <= 1) or two roots, a smaller one in [0, 1) and the trivial one at 1 (when m > 1). The theorem of Galton-Watson, made rigorous by Steffensen, is that the extinction probability is the SMALLEST root in [0, 1]. Concretely: if m <= 1 then q = 1 (extinction is certain), and if m > 1 then q < 1 is the unique root strictly below 1 (the population survives with positive probability 1 - q).
Why it matters: this is the cleanest dichotomy in all of stochastic modelling, and the smallest-root rule is exactly the same machinery that gives the survival probability of an epidemic (the basic reproduction number R_0 plays the role of m), the size of the giant component in a random graph, and the chance a nuclear chain reaction goes critical. A common error is to forget that s = 1 is ALWAYS a fixed point: the content of the theorem is choosing the smaller root, not solving q = f(q) blindly. A second subtlety: the boundary case p_1 = 1 (each individual has exactly one child) has m = 1 but the population never dies, so the careful statement excludes this degenerate offspring law from 'm <= 1 implies certain extinction'.
If each individual has offspring Poisson(m), then f(s) = exp(m(s-1)), and the extinction probability q solves q = exp(m(q-1)). For m = 2 the smallest root is q approximately 0.2032, so the lineage survives forever with probability about 0.797. For m = 1 the only root is q = 1: extinction is certain even though E[Z_n] stays at 1.
Extinction probability q is the smallest solution of q = f(q): q = 1 when m <= 1, and q < 1 when m > 1.
s = 1 is always a fixed point; the theorem is to take the SMALLEST root in [0,1]. The degenerate law p_1 = 1 has m = 1 but never dies, so it is excluded from the 'm <= 1 implies extinction' statement.