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The design of wireless sensor networks for signal processing applications has to consider the limited battery power of the sensor nodes. In this paper, a combined routing path selection and power allocation strategy is presented that is especially designed for distributed detection in sensor networks with a serial topology. The objective is to minimize the global probability of detection error of the serial network under a total network power constraint. The cross-layer approach for the selection of a routing path through the network is based on the local observation SNR of the sensors as well as fading conditions and is efficiently implemented by a backward greedy algorithm. After the routing path is established, a subsequently performed power allocation algorithm aims to optimally distribute a total power budget with respect to the detection performance of the sensor network.