Discrete two-generator subgroups of $\mathrm{P}\mathrm{S}{\mathrm{L}}_2$ over non-archimedean local fields

Let $K$ be a non-archimedean local field with residue field of characteristic $p$. We give necessary and sufficient conditions for a two-generator subgroup $G$ of $\mathrm{P}\mathrm{S}{\mathrm{L}}_2(K)$ to be discrete, where either $K={\mathbb{Q}}_p$ or $G$ contains no elements of order $p$. We give a practical algorithm to decide whether such a subgroup $G$ is discrete. We also give practical algorithms to decide whether a two-generator subgroup of either $\mathrm{S}{\mathrm{L}}_2(\mathbb{R})$ or $\mathrm{S}{\mathrm{L}}_2(K)$ (where $K$ is a finite extension of ${\mathbb{Q}}_p$) is dense. A crucial ingredient for this work is a structure theorem for two-generator groups acting by isometries on a $\mathrm{\Lambda }$-tree.