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We consider source-initiated broadcast session traffic in an ad hoc wireless network operating under a hard constraint on the end-to-end delay between the source and any node in the network. Our objective in this paper is to construct an energy-efficient broadcast tree that has a maximum depth Delta, where Delta represents the end-to-end delay constraint in the network. We first characterize the optimal solution to a closely related problem in dense networks using a dynamic programming formulation. The solution to the dynamic program indicates that there is a single optimal policy applicable to all dense networks. Elaborating on the insights provided by the structure of the problem in dense networks, we design an algorithm for finding a solution to the delay constrained minimum power broadcasting problem in sparse networks.