Distributed storage systems provide large-scale reliable data storage services by spreading redundancy across a large group of storage nodes. In such a large system, node failures take place on a regular basis. When a storage node breaks down, a replacement node is expected to regenerate the redundant data as soon as possible in order to maintain the same level of redundancy. Previous results have been mainly focused on the minimization of network traffic in regeneration. However, in practical networks, where link capacities vary in a wide range, minimizing network traffic does not always yield the minimum regeneration time. In this paper, we investigate two approaches to the problem of minimizing regeneration time in networks with heterogeneous link capacities. The first approach is to download different amounts of repair data from the helping nodes according to the link capacities. The second approach generalizes the conventional star-structured regeneration topology to tree-structured topologies so that we can utilize the links between helping nodes with bypassing low-capacity links. Simulation results show that the flexible tree-structured regeneration scheme that combines the advantages of both approaches can achieve a substantial reduction in the regeneration time.