= Finite total progeny law for binary offspring
{title2=$\mathbb P(K=2j+1)=C_jp^j(1-p)^{j+1}$}
For a <branching process> starting from one individual with offspring two with probability $p$ and zero otherwise, the finite <total progeny> is odd. Exploring individuals gives an active queue beginning at one, with increments $+1$ or $-1$. A first hit of zero at time $2j+1$ requires $j$ positive and $j+1$ negative increments and no earlier empty queue. The <Catalan number> $C_j=\binom{2j}j/(j+1)$ counts these admissible sequences, for example by the reflection principle after removing the final down step. Independence then gives the displayed mass formula.
If $p\ne1/2$, $b=4p(1-p)<1$ and $C_j\le4^j$, so $\mathbb P(K=2j+1)\le(1-p)b^j$. Choosing $c=\min\{-\log(1-p),-\tfrac12\log b\}>0$ bounds every finite mass by $e^{-c(2j+1)}$. This does not exclude an atom at infinity. In the supercritical case that atom has mass $(2p-1)/p$. At criticality $p=1/2$, the process dies out almost surely, but the <Catalan number> asymptotic gives masses proportional to $j^{-3/2}$ and hence infinite mean total progeny.
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