In a recent publication, Bruinsma has obtained a recursion relation for the random-field distribution functions in the random-field Ising model on a Bethe lattice. He has concentrated on the T=0 properties that follow from this recursion relation. We explore the finite-temperature phase diagram by analyzing the limit of the distribution averages. We obtain first- and second-order transition lines and a tricritical point. We also obtain numerically the fixed-point distributions.