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A tight $r$-tree $T$ is an $r$-uniform hypergraph that has an edge-ordering$e_1, e_2, \dots, e_t$ such that for each $i\geq 2$, $e_i$ has a vertex $v_i$that does not belong to any previous edge and $e_i-v_i$ is contained in $e_j$for some $j<i$. Kalai conjectured in 1984 that every $n$-vertex $r$-uniformhypergraph with more than $\frac{t-1}{r}\binom{n}{r-1}$ edges contains everytight $r$-tree $T$ with $t$ edges.  A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree $T'$ of $T$ such thatvertices in $V(T)\setminus V(T')$ are leaves in $T$. Kalai's Conjecture wasproved in 1987 for tight $r$-trees that have a trunk of size one. In a previouspaper we proved an asymptotic version of Kalai's Conjecture for all tight$r$-trees that have a trunk of bounded size. In this paper we continue thatwork to establish the exact form of Kalai's Conjecture for all tight $3$-treeswith at least $20$ edges that have a trunk of size two.

Hypergraphs not containing a tight tree with a bounded trunk ~II: 3-trees with a trunk of size 2