We establish an unusual 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 [7]. 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 (from the point of view
of convergence in probability), already described by Derrida and Spohn [14].
Surprisingly, this phase transition phenomenon disappears in the sense of upper
almost sure limits.