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A Galton-Watson process is a branching process that counts individuals by generation. Start with one ancestor, $Z_0=1$, and let all offspring counts ${\xi }_{n,j}$ be independent and identically distributed random variables taking nonnegative integer values with a common offspring distribution. ThenThe empty sum is zero: zero is an absorbing state, so once a generation is empty every later generation is empty. The same offspring distribution applies to every individual in every generation. ^[ref]

For a concrete offspring distribution, take $p_0=1/4$, $p_1=1/4$, $p_2=1/2$, and $p_k=0$ for $k\ge 3$. An individual has no children with probability $1/4$, one with probability $1/4$, and two with probability $1/2$. The offspring expected value is $\mu =5/4$. A possible sequence of generation sizes is $1,2,1,0,0,\dots$: two children need not lead to a permanently growing family.

The probability generating function $F(s)=1/4+s/4+s^2/2$ gives the exact generation probability distributions through branching-process generating-function iteration. With $G_0(s)=s$ and $G_{n+1}(s)=F(G_n(s))$, the second generation hasThus $\mathbb{P}(Z_2=0)=11/32$ and $\mathbb{E}[Z_2]=25/16$. The figure shows full generation probability mass functions, computed by polynomial function composition, through generation six; no simulated frequencies are used.

The contrast is striking: the expected generation size in a Galton-Watson process grows as $(5/4{)}^n$, while the branching extinction probability is $1/2$. By generation six, $\mathbb{P}(Z_6=0)\approx 0.462186$ although $\mathbb{E}[Z_6]\approx 3.814697$. Larger surviving families carry the rising expected value; that mean is an average over possible families, not a forecast of one family's path. The Galton-Watson extinction fixed point turns an unbounded number of future generations into a single equation for this probability.

The historical question was the disappearance of surnames passed through male descendants. In Watson and Galton, On the Probability of the Extinction of Families, pages 139–140, Francis Galton formulates a constant-law model counting sons who reach adulthood, and Watson computes later generation probability distributions by composing generating functions. Those are assumptions of their model, rather than a claim that actual human reproduction follows this offspring distribution.

The offspring probabilities $P(K=j)=2^{-j-1}$ for $j\ge 0$ have expected value one. For one initial individual, the branching-process generating-function iteration givesFor $n\ge 1$ the PGF disk is $|s|<1+1/n$; $G_0(s)=s$ separately. Conditional on survival, the population is positive geometric with success probability $1/(n+1)$, leading to the exponential limit of surviving geometric branching.

Condition on the current generation size. By independence of offspring families, $\mathbb{E}[s^{Z_{n+1}}\mid Z_n=k]=G(s{)}^k$. Taking the expected value gives the probability generating function recursionInduction therefore proves the branching-process generating-function iterationwhere the right side means $n$-fold composition, not an ordinary power.

Let $m=G^{\prime }(1)$ and ${\sigma }^2=G^{\prime \prime }(1)+m-m^2$ be the offspring mean and variance, assuming a finite second moment. Differentiating the generating-function recursion at one gives $\mathbb{E}{Z}_n=m^n$ and, with $a_n=G_n^{\prime \prime }(1)$,Since $\mathrm{Var}(Z_n)=a_n+m^n-m^{2n}$, this impliesIteration and summation of the geometric progression give the mean and variance of a Galton-Watson generation:For $m=0$ the offspring count is identically zero, so every later generation has zero variance. The zeroth generation always has zero variance. Finiteness of the needed moments is an assumption for this finite-valued variance formula, not a consequence of merely having a probability generating function. With finite positive mean but infinite offspring variance, all positive-generation variances are infinite.

For the total count, make the generation convention explicit. If the first $n$ generations include the ancestor, put $T_n=Z_0+\cdots +Z_{n-1}$ and $H_n(s)=\mathbb{E}[s^{T_n}]$, with $T_0=0$. Given $k$ children of the ancestor, each starts an independent depth-$n$ descendant tree with generating function $H_n$. The ancestor contributes an additional factor $s$, soThis is the finite-horizon total progeny generating function. If instead “first $n$ generations” means descendants $Z_1+\cdots +Z_n$, excluding $Z_0$, write ${\hat{H}}_n$ for that convention. Conditioning on the children now gives ${\hat{H}}_{n+1}(s)=G(s{\hat{H}}_n(s))$, with ${\hat{H}}_0=1$ and ${\hat{H}}_1=G$. The two conventions encode the same tree but different counts.

Conditional on $X_n=m$, the next generation is the sum of $m$ independent offspring counts, each with probability generating function $F_1(s)$. HenceThis also holds for $m=0$, when the empty sum is zero and the empty product is one. The law of total expectation givesFor complex $|s|\le 1$, the expectations are absolutely bounded because $|F_1(s)|\le 1$. This proves the branching-process generating-function iteration, with $F_0(s)=s$.

For a Galton-Watson process, condition on the number $K$ 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 givesThis 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 $|s|\le 1$, with larger domains available when the corresponding expectations converge.