  We provide sufficient criteria for explosion in Crump-Mode-Jagers branching
process, via the process producing an infinite path in finite time. As an
application, we deduce a curious phase-transition in the infinite tree
associated with a class of recursive tree models with fitness, showing that in
one regime every node in the tree has infinite degree, whilst in another, the
tree is locally finite, with a unique infinite path. The latter class
encompasses many models studied in the literature, including the weighted
random recursive tree, the preferential attachment tree with additive fitness,
and the Bianconi-Barab\'asi model, or preferential attachment tree with
multiplicative fitness.
