  Vine copulas can efficiently model multivariate probability distributions.
This paper focuses on a more thorough understanding of their structures, since
in the literature, vine copula representations are often ambiguous. The graph
representations include the original, cherry and chordal graph sequence
structures, which we show equivalence between. Importantly we also show a new
result, namely that when a perfect elimination ordering of a vine structure is
given, then it can always be uniquely represented with a matrix. O. M.
N\'apoles has shown a way to represent vines in a matrix, and we algorithmify
this previous approach, while also showing a new method for constructing such a
matrix, through cherry tree sequences. We also calculate the runtime of these
algorithms. Lastly, we prove that these two matrix-building algorithms are
equivalent if the same perfect elimination ordering is being used.
