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We consider the problem of maximizing the time-to-first-failure (TTFF), defined as the time till the first node in the network runs out of battery energy, in energy constrained broadcast wireless networks. We show that the TTFF criterion, by itself, fails to provide the "ideally optimum" multicast tree and propose a composite weighted objective function which maximizes the TTFF and minimizes the sum of transmitter powers. We then develop a mixed integer linear programming (MILP) model for solving the joint optimization problem optimally. We also consider the case of prioritized nodes and show how the model can be modified to deal with such priorities.