In this paper, we study the problem of routing, bandwidth and flow assignment in Wireless Body Area Networks (BANs). Our solution considers BAN for real-time data streaming applications, where the real-time nature of data streams is of critical importance for providing a useful and efficient sensorial feedback for the user while system lifetime should be maximized. Thus, bandwidth and energy efficiency of the communication between energy constrained sensor nodes must be carefully optimized. The proposed solution takes into account nodes' residual energy during the establishment of the routing paths and adaptively allocates bandwidth to the nodes in the network. We also formulate the joint routing tree construction and bandwidth allocation problem as an Integer Linear Program that maximizes the network utility while satisfying the QoS requirements. We compare the resulting performance of our protocol with the optimal solution, and show that it closes a considerable portion of the gap from the theoretical optimal solution.