= Solution
Count the original infected person first. Conditional on their producing $X=k$ new infections, each of the $k$ new people initiates an independent descendant family with the same law as the original total. If $T_i$ denotes that family size, the branching property gives
$$
\boxed{T\overset d=1+\sum_{i=1}^{X}T_i.}
$$
The family variables are mutually independent and independent of $X$. At $X=0$ the sum is empty and equals zero, so the total is one. This is the total progeny decomposition of a <Galton-Watson process>.
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