  The present paper explores a connection between two concepts arising from
different fields of mathematics. The first concept, called vine, is a graphical
model for dependent random variables. This concept first appeared in a work of
Joe (1994), and the formal definition was given later by Cooke (1997). Vines
have nowadays become an active research area whose applications can be found in
probability theory and uncertainty analysis. The second concept, called
MAT-freeness, is a combinatorial property in the theory of freeness of
logarithmic derivation modules of hyperplane arrangements. This concept was
first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards
investigated further by Cuntz-M{\"u}cksch (2020).
  In the particular case of graphic arrangements, the last two authors (2023)
recently proved that the MAT-freeness is completely characterized by the
existence of certain edge-labeled graphs, called MAT-labeled graphs. In this
paper, we first introduce a poset characterization of a vine, the so-called
vine. Then we show that, interestingly, there exists an explicit equivalence
between the categories of locally regular vines and MAT-labeled graphs. In
particular, we obtain an equivalence between the categories of regular vines
and MAT-labeled complete graphs.
  Several applications will be mentioned to illustrate the interaction between
the two concepts. Notably, we give an affirmative answer to a question of
Cuntz-M{\"u}cksch that MAT-freeness can be characterized by a generalization of
the root poset in the case of graphic arrangements.
