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Achievable energy conservation rate in wireless sensor networks can further be enhanced depending on application-specific requirements such as data reporting latency and sensing coverage area. In this paper, we propose a novel framework for energy-conserving data gathering which exploits a trade-off between coverage and data reporting latency. The ultimate goal is to extend the network lifetime by offering only the application/user-specific quality of service (e.g., sensing coverage) in each data reporting round using (approximately) k sensors. The selection of these k sensors is based on a geometric probability theory and a randomization technique with constant computational complexity. The selected k sensors form a data gathering tree (DGT) rooted at the data gathering point. To improve on energy-savings, only the sensors on the DGT are scheduled to remain active (with transceiver on) during a reporting round so that the DGT is used as a backbone to reach the data gathering point. We present a probabilistic model for estimating the connectivity of the selected k sensors and also a recursive algorithm which derives the number of additional sensors required to probabilistically guarantee the connectivity. The immediate data reporting capability of sensors is also analyzed since the group of active sensors forming a DGT (backbone) in each round may not be able to serve as a dominating set. Simulation results show that the proposed framework leads to a significant conservation of energy with a small trade-off.