Ancestral reproductive bias in continuous-time branching trees under various sampling schemes

Cheek and Johnston (JMB 86:70, 2023) consider a continuous-time Bienaymé-Galton-Watson tree conditioned on being alive at time T . They study the reproduction events along the ancestral lineage of an individual randomly sampled from all those alive at time T . We give a short proof of an extension of their main results (Cheek and Johnston in JMB 86:70, 2023, Theorems 2.3 and 2.4) to the more general case of Bellman-Harris processes. Our proof also sheds light onto the probabilistic structure of the rate of the reproduction events. A similar method will be applied to explain (i) the different ancestral reproduction bias appearing in work by Geiger (JAP 36:301–309, 1999) and (ii) the fact that the sampling rule considered by Chauvin et al. (SPA 39:117–130, 1991), (Theorem 1) leads to a time homogeneous process along the ancestral lineage.