  Let $T$ be an arbitrary phylogenetic tree with $n$ leaves. It is well-known
that the average quartet distance between two assignments of taxa to the leaves
of $T$ is $\frac 23 \binom{n}{4}$. However, a longstanding conjecture of
Bandelt and Dress asserts that $(\frac 23 +o(1))\binom{n}{4}$ is also the {\em
maximum} quartet distance between two assignments. While Alon, Naves, and
Sudakov have shown this indeed holds for caterpillar trees, the general case of
the conjecture is still unresolved. A natural extension is when partial
information is given: the two assignments are known to coincide on a given
subset of taxa. The partial information setting is biologically relevant as the
location of some taxa (species) in the phylogenetic tree may be known, and for
other taxa it might not be known. What can we then say about the average and
maximum quartet distance in this more general setting? Surprisingly, even
determining the {\em average} quartet distance becomes a nontrivial task in the
partial information setting and determining the maximum quartet distance is
even more challenging, as these turn out to be dependent of the structure of
$T$. In this paper we prove nontrivial asymptotic bounds that are sometimes
tight for the average quartet distance in the partial information setting. We
also show that the Bandelt and Dress conjecture does not generally hold under
the partial information setting. Specifically, we prove that there are cases
where the average and maximum quartet distance substantially differ.
