Forests of Gaussian numbers

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.