Yes, that's the Stern-Brocot tree discussed in the same paper.  The trouble
is that you need to keep around more (indeed, increasingly more) state.
Here's the Haskell version from the paper:

rats :: [Rational]
rats = concat (unfolds infill [(0,1),(1,0)])
unfolds f a = let (b,a') = f a in b : unfolds f a0
infill xs = (map mkRat ys,interleave xs ys)
                 where ys = zipWith adj xs (tail xs)
interleave (x : xs) ys = x : interleave ys xs
interleave [] [] = []