  We consider a model of directed polymers on a regular tree with a disorder
given by independent, identically distributed weights attached to the vertices.
For suitable weight distributions this model undergoes a phase transition with
respect to its localization behaviour. We show that, for high temperatures, the
free energy is supported by a random tree of positive exponential growth rate,
which is strictly smaller than that of the full tree. The growth rate of the
minimal supporting subtree is decreasing to zero as the temperature decreases
to the critical value. At the critical value and all lower temperatures, a
single polymer suffices to support the free energy. Our proofs rely on elegant
martingale methods adapted from the theory of branching random walks.
