  Let $\mathcal{D}_0:= \{x + iy \ \vert x, y >0\}$, and let $(L, R)$ be a pair
of M\"{o}bius transformations corresponding to $\mathrm{SL}_2(\mathbb{N}_0)$
matrices such that $R(\mathcal{D}_0)$ and $L(\mathcal{D}_0)$ are disjoint.
Given such a pair (called a left-right pair), we can construct a directed graph
$\mathcal{F}(L, R)$ with vertices $\mathcal{D}_0$ and edges $\{(z, R(z))\}_{z
\in \mathcal{D}_0} \cup \{(z, L(z))\}_{z \in \mathcal{D}_0}$, which is a
collection of infinite binary trees. We answer two questions of Nathanson by
classifying all the pairs of elements of $\mathrm{SL}_2(\mathbb{N}_0)$ whose
corresponding M\"{o}bius transformations form left-right pairs and showing that
trees in $\mathcal{F}(L, R)$ are always rooted.
