Denominators in grouped birth-outcome models

For each treatment year let $C$ be cycles, $S$ singleton deliveries, $T$ twin deliveries and $H$ deliveries of three or more babies. A singleton-per-cycle model uses $S\sim \mathrm{Bin}(C,p)$. A multiple-delivery model conditional on a live delivery instead uses $T+H\sim \mathrm{Bin}(S+T+H,q)$. The latter denominator counts deliveries, not babies, and it excludes failed cycles. Confusing these denominators changes the statistical question.