  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 [9]. 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 [17].
Surprisingly, this phase transition phenomenon disappears in the sense of upper
almost sure limits.
