K

of children of the initial individual. The descendants of these children over the next

n

generations are independent copies of

X_{n}

. The probability generating function of their total, conditional on

K = k

, is

G_{n} (s)^{k}

. Averaging over the family-size law gives

G_{n + 1} (s) = E [G_{n} (s)^{K}] = G (G_{n} (s)), G_{0} (s) = s .

This is the branching-process generating-function iteration. Conditioning instead on the last generation gives

G_{n} (G (s))

; both orders agree because

G_{n}

is the

n

th iterate of the same function

G

. The argument holds on the usual PGF domain

∣ s ∣ \leq 1

, with larger domains available when the corresponding expectations converge.

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