  We study the two-dimensional continued fraction algorithm introduced in
\cite{garr} and the associated \emph{triangle map} $T$, defined on a triangle
$\triangle\subset \R^2$. We introduce a slow version of the triangle map, the
map $S$, which is ergodic with respect to the Lebesgue measure and preserves an
infinite Lebesgue-absolutely continuous invariant measure. We discuss the
properties that the two maps $T$ and $S$ share with the classical Gauss and
Farey maps on the interval, including an analogue of the weak law of large
numbers and of Khinchin's weak law for the digits of the triangle sequence, the
expansion associated to $T$. Finally, we confirm the role of the map $S$ as a
two-dimensional version of the Farey map by introducing a complete tree of
rational pairs, constructed using the inverse branches of $S$, in the same way
as the Farey tree is generated by the Farey map, and then, equivalently,
generated by a generalised mediant operation.
