An article in the new April issue of Notices of the American
Mathmatical Society suggests a search for connections
between the Calkin-Wilf tree and the modular group.
The search yields, for instance (in chronological order) …
"Cutting sequences for geodesic flow on the modular surface
and continued fractions," David J. Grahinet, Jeffrey C. Lagaria,
arXiv, 2 April 2001
"Orderings of the rationals and dynamical systems,"
Claudio Bonanno, Stefano Isola, arXiv, 14 May 2008.
"Periods of negative-regular continued fractions. Rational numbers."
Sergey Khrushchev and Michael Tyaglov, slides PDF, 11 Sept. 2012
"The Minkowski ?(x) function, a class of singular measures,
theta-constants, and mean-modular forms," Giedrius Alkauskas,
arXiv, 20 Sept. 2012
"Forests of complex numbers,"
Melvyn B. Nathanson, arXiv, 1 Dec. 2014
Update of March 21, 2015:
For many more related papers, search by combining the
phrase "modular group" with phrases denoting forking structures
other than Calkin-Wilf, such as "cubic tree," "Stern-Brocot tree,"
and "Farey tree" (or "Farey sequence" or "Farey series" or
"Farey graph" ).