 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 givesThe 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.