  We consider Metropolis Glauber dynamics for sampling proper $q$-colourings of
the $n$-vertex complete $b$-ary tree when $3\leq q\leq b/2\ln(b)$. We give both
upper and lower bounds on the mixing time. For fixed $q$ and $b$, our upper
bound is $n^{O(b/\log b)}$ and our lower bound is $n^{\Omega(b/q \log(b))}$,
where the constants implicit in the $O()$ and $\Omega()$ notation do not depend
upon $n$, $q$ or $b$.
