In this paper, we present a new detector for the detection of sparse stochastic signals using 1-bit data in two-level two-degree tree-structured sensor networks (2L-2D TSNs). Related prior work mostly concentrates on parallel sensor networks (PSNs). However, PSNs may sometime become impractical in many applications including the case where some sensors are beyond the communication range of the fusion center (FC). Therefore, we design the proposed detector for 2L-2D TSNs where information is transmitted hierarchically. To satisfy severe resource constraints, each local sensor performs 1-bit quantization before transmission to the FC. The FC fuses the received 1-bit data employing the locally most powerful test (LMPT). It is shown theoretically and numerically that, compared with the LMPT detector with Q sensors that transmit analog measurements in 2L-2D TSNs, the proposed 1-bit LMPT detector that uses quantization thresholds derived in this paper asymptotically requires 1.74Q sensors to compensate for the performance loss induced by local quantization.