We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou (2005). Our method applies furthermore to the study of directed polymers on a disordered tree; in particular, we give a rigorous proof of a phase transition phenomenon for the partition function, described by Derrida and Spohn (1988).