Let T be an infinite rooted tree with weights we assigned to its edges. Denote by mn(T) the minimum weight of a path from the root to a node of the nth generation. We consider the possible behaviour of mn(T) with focus on the two following cases: we say T is explosive if limn→∞mn(T)<∞,and say that T exhibits linear growth if lim infn→∞mn(T)n>0.We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres (Ann Probab 22(1), 180–194, 1994). As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.