Based on thework of Durhuus–Jónsson 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 this 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_*.

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