  Finding optimal evolutionary trees from sequence data is typically an
intractable problem, and there is usually no way of knowing how close to
optimal the best tree from some search truly is. The problem would seem to be
particularly acute when we have many taxa and when that data has high levels of
homoplasy, in which the individual characters require many changes to fit on
the best tree. However, a recent mathematical result has provided a precise
tool to generate a short number of high-homoplasy characters for any given
tree, so that this tree is provably the optimal tree under the maximum
parsimony criterion. This provides, for the first time, a rigorous way to test
tree search algorithms on homoplasy-rich data, where we know in advance what
the `best' tree is. In this short note we consider just one search program
(TNT) but show that it is able to locate the globally optimal tree correctly
for 32,768 taxa, even though the characters in the dataset requires, on
average, 1148 state-changes each to fit on this tree, and the number of
characters is only 57.
