  The closure $\textrm{cl}(R)$ of a consistent set $R$ of triples (rooted
binary trees on three leaves) provides essential information about tree-like
relations that are shown by any supertree that displays all triples in $R$. In
this contribution, we are concerned with representative triple sets, that is,
subsets $R'$ of $R$ with $\textrm{cl}(R') = \textrm{cl}(R)$. In this case, $R'$
still contains all information on the tree structure implied by $R$, although
$R'$ might be significantly smaller. We show that representative triple sets
that are minimal w.r.t.\ inclusion form the basis of a matroid. This in turn
implies that minimal representative triple sets also have minimum cardinality.
In particular, the matroid structure can be used to show that minimum
representative triple sets can be computed in polynomial time with a simple
greedy approach. For a given triple set $R$ that "identifies" a tree, we
provide an exact value for the cardinality of its minimum representative triple
sets. In addition, we utilize the latter results to provide a novel and
efficient method to compute the closure $\textrm{cl}(R)$ of a consistent triple
set $R$ that improves the time complexity $\mathcal{O}(|R||L_R|^4)$ of the
currently fastest known method proposed by Bryant and Steel (1995). In
particular, if a minimum representative triple set for $R$ is given, it can be
shown that the time complexity to compute $\textrm{cl}(R)$ can be improved by a
factor up to $|R||L_R|$. As it turns out, collections of quartets (unrooted
binary trees on four leaves) do not provide a matroid structure, in general.
