We propose two novel types of spatially multiplexed single-photon sources based on incomplete binary-tree multiplexers. The incomplete multiplexers are extensions of complete binary-tree multiplexers, and they contain incomplete branches either at the input or at the output of them. We analyze and optimize these systems realized with general asymmetric routers and photon-number-resolving detectors by applying a general statistical theory introduced previously that includes all relevant loss mechanisms. We show that the use of any of the two proposed multiplexing systems can lead to higher single-photon probabilities than that achieved with complete binary-tree multiplexers. Single-photon sources based on output-extended incomplete binary-tree multiplexers outperform those based on input-extended ones in the considered parameter ranges, and they can in principle yield single-photon probabilities higher than 0.93 when they are realized by state-of-the-art bulk optical elements. We show that the application of the incomplete binary-tree approach can significantly improve the performance of the multiplexed single-photon sources for suboptimal system sizes that is a typical situation in current experiments.