An important ingredient in the asymptotic safety scenario for quantum gravity is the existence of a suitable point in theory space modeling quantum geometry with exact scaling symmetry. It is natural to look for candidates in scaling limits of random discrete geometries, like the random triangulated spheres featuring in Euclidean Dynamical Triangulations (EDT). However, in dimensions higher than two, there are serious mathematical challenges in studying these models analytically, while numerical simulations are yet to uncover promising critical phenomena in these systems. I will discuss recent joint work (arxiv:2203.16105) with Luca Lionni, in which we considered a restricted family of triangulations of the 3-sphere that can be encoded in certain trees. These triangulations are under better enumerative control (they are locally constructible and admit explicit exponential bounds) and exploratory numerical simulations point at qualitative differences with vanilla EDT.