  A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex
set into `bags', such that identifying the vertices in each bag produces a
forest. The \emph{tree-partition-width} of $G$ is the minimum number of
vertices in a bag in a tree-partition of $G$. An anonymous referee of the paper
by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph
with tree-width $k\geq3$ and maximum degree $\Delta\geq1$ has
tree-partition-width at most $24k\Delta$. We prove that this bound is within a
constant factor of optimal. In particular, for all $k\geq3$ and for all
sufficiently large $\Delta$, we construct a graph with tree-width $k$, maximum
degree $\Delta$, and tree-partition-width at least $(\eighth-\epsilon)k\Delta$.
Moreover, we slightly improve the upper bound to ${5/2}(k+1)({7/2}\Delta-1)$
without the restriction that $k\geq3$.
