Linkages between the Gauss map and the Stern-Brocot tree

We discover a bijective map between the Gauss Map and the left-half of the Stern-Brocot Tree. The domain of the Gauss Map is then extended to cover all reals, and the coverage of the Stern-Brocot Tree is extended to include all positive and negative rationals in a manner that preserves the map between the two constructions.