On the structure of genealogical trees associated with explosive Crump-Mode-Jagers branching processes

We study the structure of genealogical trees associated with explosive Crump-Mode-Jagers branching processes (stopped at the explosion time), proving criteria for the associated tree to contain a node of infinite degree (a star) or an infinite path. Next, we provide uniqueness criteria under which with probability $1$ there exists exactly one of a unique star or a unique infinite path. Under the latter uniqueness criteria, we also provide an example where, with strictly positive probability less than $1$, there exists a unique node of infinite degree in the model, thus this probability is not restricted to being $0$ or $1$. Moreover, we provide structure theorems when there is a star, when certain trees appear as sub-trees of the star infinitely often. We apply our results to general discrete evolving tree models of explosive recursive trees with fitness, and as particular cases, we study a family of super-linear preferential attachment models with fitness. In the latter regime, we derive phase transitions in the model parameters in three different examples, leading to either exactly one star with probability $1$, or one infinite path with probability $1$, with every node having finite degree. Furthermore, we highlight examples where sub-trees $T$ of arbitrary size can appear infinitely often; behaviour that is markedly distinct from super-linear preferential attachment models studied in the literature so far.