  It is a classical result that an unrooted tree $T$ having positive
real-valued edge lengths and no vertices of degree two can be reconstructed
from the induced distance between each pair of leaves. Moreover, if each
non-leaf vertex of $T$ has degree 3 then the number of distance values required
is linear in the number of leaves. A canonical candidate for such a set of
pairs of leaves in $T$ is the following: for each non-leaf vertex $v$, choose a
leaf in each of the three components of $T-v$, group these three leaves into
three pairs, and take the union of this set over all choices of $v$. This forms
a so-called 'triplet cover' for $T$. In the first part of this paper we answer
an open question (from 2012) by showing that the induced leaf-to-leaf distances
for any triplet cover for $T$ uniquely determine $T$ and its edge lengths. We
then investigate the finer combinatorial properties of triplet covers. In
particular, we describe the structure of triplet covers that satisfy one or
more of the following properties of being minimal, 'sparse', and 'shellable'.
