  The search for universality in random triangulations of manifolds, like those
featuring in (Euclidean) Dynamical Triangulations, is central to the random
geometry approach to quantum gravity. In case of the 3-sphere, or any other
manifold of dimension greater than two for that matter, the pursuit is held
back by serious challenges, including the wide open problem of enumerating
triangulations. In an attempt to bypass the toughest challenges we identify a
restricted family of triangulations, of which the enumeration appears less
daunting. In a nutshell, the family consists of triangulated 3-spheres
decorated with a pair of trees, one spanning its tetrahedra and the other its
vertices, with the requirement that after removal of both trees one is left
with a tree-like 2-complex. We prove that these are in bijection with a
combinatorial family of triples of plane trees, satisfying restrictions that
can be succinctly formulated at the level of planar maps. An important
ingredient in the bijection is a step-by-step reconstruction of the
triangulations from triples of trees, that results in a natural subset of the
so-called locally constructible triangulations, for which spherical topology is
guaranteed, through a restriction of the allowed moves. We also provide an
alternative characterization of the family in the framework of discrete Morse
gradients. Finally, several exponential enumerative bounds are deduced from the
triples of trees and some simulation results are presented.
