Superconducting nanowire avalanche single-photon detectors (SNAPs) with n parallel nanowires are advantageous over single-nanowire detectors because their output signal amplitude scales linearly with n. However, the SNAP architecture has not been viably demonstrated for n > 4. To increase n for larger signal amplification, we designed a multi-stage, successive-avalanche architecture which used nanowires, connected via choke inductors in a binary-tree layout. We demonstrated an avalanche detector with n = 8 parallel nanowires and achieved eight-fold signal amplification, with a timing jitter of 54 ps.