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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.