  The Farey sequence $\mathcal{F}(Q)$ at level $Q$ is the sequence of
irreducible fractions in $[0, 1]$ with denominators not exceeding $Q$, arranged
in increasing order of magnitude. A simple ``next-term'' algorithm exists for
generating the elements of $\mathcal{F}(Q)$ in increasing or decreasing order.
That algorithm, along with a number of other properties of the Farey sequence,
was encoded by F. Boca, C. Cobeli, and A. Zaharescu into what is now known as
the Boca-Cobeli-Zaharescu (BCZ) map, and used to attack several problems that
can be described using the statistics of subsets of the Farey sequence. In this
paper, we derive the Boca-Cobeli-Zaharescu map analogue for the discrete orbits
$\Lambda_q = G_q(1, 0)^T$ of the linear action of the Hecke triangle groups
$G_q$ on the plane $\mathbb{R}^2$ starting with a Stern-Brocot tree analogue
for the said orbits. We derive the next-term algorithm for generating the
elements of $\Lambda_q$ in vertical strips in increasing order of slope, and
present a number of applications to the statistics of $\Lambda_q$.
