In this paper, we propose a novel heuristic algorithm for constructing a minimum cost multicast tree. Our work is based on a directed asymmetric network and shows an improvement in terms of network cost for general random topologies close to real networks. It is compared to the most effective scheme proposed earlier by Takahashi and Matsuyama (TM) . We have experimented comprehensive computer simulations and the performance enhancement is up to about 4.7% over TM. The time complexity of ours is
) for an
-node network with
members in the multicast group which is comparable to those of previous works [12,18].