  We prove that for every tree $T$ of radius $h$, there is an integer $c$ such
that every $T$-minor-free graph is contained in $H\boxtimes K_c$ for some graph
$H$ with pathwidth at most $2h-1$. This is a qualitative strengthening of the
Excluded Tree Minor Theorem of Robertson and Seymour (GM I). We show that
radius is the right parameter to consider in this setting, and $2h-1$ is the
best possible bound.
