  The Calkin-Wilf tree is an infinite binary tree whose vertices are the
positive rational numbers. Each number occurs in the tree exactly once and in
the form $a/b$, where are $a$ and $b$ are relatively prime positive integers.
In this paper, certain subsemigroups of the modular group are used to construct
similar trees in the set $\mathcal{D}_0$ of positive complex numbers.
Associated to each semigroup is a forest of trees that partitions
$\mathcal{D}_0$. The set of cusps and the fundamental domain of the semigroup
are defined and computed.
