  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 system 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.
