Regular trees of graphs are inverse limits of particularly simple inverse systems of finite graphs. They form a 1-dimensional subclass of the Markov compacta: a class of finitely describable inverse limits of simplicial complexes, which includes all boundaries of hyperbolic groups. I will discuss upcoming joint work with Jacek Swiatkowski in which we use Bowditch's canonical JSJ decomposition to characterize the 1-ended hyperbolic groups whose boundaries are (regular) trees of graphs.