We revisit the problem of counting paths in width-2 planar branching programs. We show that this is hard for Boolean NC1 under ACC 0[5] reductions, completing a proof strategy outlined in [3]. On the other hand, for several restricted instances of width-2 planar branching programs, we show that the counting problem is TC 0-complete. We also show that non-planar width-2 programs can be planarized in AC 0[2]. Using the equivalence of planar width-2 programs with the reduced-form representation of positive rationals, we show that the evaluation problem for this representation in the Stern-Brocot tree is also NC 1 hard. In contrast, the evaluation problem in the continued fraction representation is in TC 0. © 2012, Australian Computer Society, Inc.