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In this paper, a distributed algorithm for the construction of load balanced routing trees in wireless sensor networks is proposed with the goal of prolonging network lifetime. The proposed method, called UDBC, achieves load balancing by adjusting the number of children among parents as much as possible. Child adjustment is considered as a game between parents and child nodes by defining their utility functions. At the game termination, each individual node gains the maximum benefit based on its utility function, and the network as a whole reaches the global goal of forming the balanced tree. UDBC generates more balanced trees compared to similar work in the literature by imposing less time and communication overheads, which is proved by analytical evaluations and simulation results.