= Solution
The ancestor contributes one individual. Given its offspring count $N$, the descendant family rooted at each child is an independent <branching process> with the same <offspring distribution> as the original one. Its <total progeny> therefore has the same distribution as $Z$. The independent reproduction choices let these descendant <total progeny> variables be represented by independent copies $Z_1,Z_2,\ldots$, independent of $N$. Hence, on this family-tree construction,
$$
\boxed{Z=1+\sum_{i=1}^N Z_i,}
$$
where the sum is zero when $N=0$.
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