It is important to design an energy efficient data gathering tree structure for wireless sensor networks. As for the energy efficiency, network’s overall energy consumption and pernode fairness have been studied. Note that the minimum degree spanning tree (MDST) is optimal in the sense of per-node fairness. We believe that the per-node fairness is of practical interest and that the delay bound associated with it must be investigated. Unfortunately, no such efforts have been made so far.In this paper, we propose a tree structure, called the DBMDST, which modifies MDST to reduce the height and extend its degree. With the DB-MDST, the near optimal delay bound (i.e., comparable to the one obtained using the shortest path tree) can be achieved at the cost of as much energy as consumed by the MDST while it cannot be achieved with the MDST. Simulation results support the claim stated.