Tur\'an numbers of hypergraph trees

Understand

An $r$-graph is an $r$-uniform hypergraph tree (or $r$-tree) if its edges can be ordered as $E_1,\ldots, E_m$ such that $\forall i>1 \, \exists \alpha(i)<i$ such that $E_i\cap (\bigcup_{j=1}^{i-1} E_j)\subseteq E_{\alpha(i)}$.

• The Tur\'an number $ex(n,{\cal H})$ of an $r$-graph ${\cal H}$ is the largest size of an $n$-vertex $r$-graph that does not contain ${\cal H}$.
• A cross-cut of ${\cal H}$ is a set of vertices in ${\cal H}$ that contains exactly one vertex of each edge of ${\cal H}$.
• The cross-cut number $\sigma({\cal H})$ of ${\cal H}$ is the minimum size of a cross-cut of ${\cal H}$.