the subcritical, critical, and supercritical classification
Every Galton-Watson process falls into exactly one of three regimes according to where the mean offspring number m sits relative to 1. This trichotomy — subcritical (m < 1), critical (m = 1), supercritical (m > 1) — is the organizing principle of the whole subject, and it is remarkable that one number controls qualitatively different long-run behaviour: certain quick death, certain but slow death, or possible explosion.
Subcritical (m < 1): the population shrinks in mean, E[Z_n] = m^n -> 0, extinction is certain (q = 1), and the time to extinction has an exponential-type tail; conditioned on survival to generation n, the population size has a limiting distribution (this is the subcritical Yaglom theorem), and the survival probability decays like m^n. Critical (m = 1, excluding p_1 = 1): the most delicate case — E[Z_n] = 1 for all n, yet extinction is still certain (q = 1). Survival is borderline: P(Z_n > 0) is asymptotic to 2/(sigma^2 n) where sigma^2 is the offspring variance, and conditioned on survival, Z_n / n converges to an exponential law (Yaglom's theorem). Supercritical (m > 1): now q < 1, so the population survives with positive probability 1 - q, and on the survival event it grows geometrically: Z_n / m^n converges almost surely to a nonnegative limit W, with W > 0 exactly on survival under the Kesten-Stigum (x log x) condition.
Why it matters: the same classification reappears everywhere a reproduction-or-decay mechanism is in play. An epidemic with reproduction number R_0 < 1 dies out (subcritical), R_0 = 1 is the epidemic threshold (critical), and R_0 > 1 permits a major outbreak (supercritical). The Erdos-Renyi giant component appears exactly when the local branching mean crosses 1. The crucial honesty here is the critical case: people are tempted to think 'mean stays constant, so it survives,' but criticality is the knife-edge where the mean is conserved yet extinction is certain — the prototype of a martingale that converges to zero. Mean behaviour and almost-sure behaviour part ways most violently exactly at m = 1.
Take binary splitting: offspring is 0 or 2 with probabilities (1-p, p), so m = 2p. If p = 0.4 then m = 0.8 < 1, subcritical, certain extinction. If p = 0.5 then m = 1, critical, still certain extinction but P(Z_n > 0) decays only like 1/n. If p = 0.6 then m = 1.2 > 1, supercritical, and the lineage survives with probability 1 - q > 0 where q solves q = (1-p) + p q^2.
One number m decides the fate: m<1 quick death, m=1 borderline (slow but certain death), m>1 possible survival.
The critical case m = 1 is the trap: people assume constant mean implies survival, but the population dies out with probability one (excluding the trivial p_1 = 1). Criticality is exactly where E[Z_n] = 1 yet Z_n -> 0 almost surely.