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Sensor nodes age over time due to device failure and/or battery energy depletion. Node survival rates affect data communication, sensing coverage, and, especially, the connectivity of data gathering trees that provide a forwarding path from each source to the sink and enable data aggregation. The node aging effect on the connectivity of a data gathering tree over time is analyzed. First, we discuss the general node aging problem by considering the device failure rate and the energy consumption rate. In the analysis of the energy consumption rate, we examine the effect of the data aggregation degree and the hop distance on the amount of data communication handled by a node of a data gathering tree. Then, we present the survival function and the connectivity probability per hop distance in a data gathering tree. Finally, the resulting non-uniform connectivity over time in a data gathering tree is examined with the comparison between the device failure effect and the energy depletion effect through extensive simulation. It is shown by mathematical analysis, as well as simulation, that the node aging process has a significant impact on connectivity as the hop distance increases.