Trees of nuclei and bounds on the number of triangulations of the 3-ball

Based on the work of Durhuus-J{\'o}nsson and Benedetti-Ziegler, we revisit the question of the number of triangulations of the 3-ball. We introduce a notion of nucleus (a triangulation of the 3-ball without internal nodes, and with each internal face having at most 1 external edge). We show that every triangulation can be built from trees of nuclei. This leads to a new reformulation of Gromov's question: We show that if the number of rooted nuclei with $t$ tetrahedra has a bound of the form $C^t$, then the number of rooted triangulations with $t$ tetrahedra is bounded by $C_*^t$.