  The Gy\'arf\'as-Sumner conjecture says that for every forest $H$ and every
integer $k$, if $G$ is $H$-free and does not contain a clique on $k$ vertices
then it has bounded chromatic number. (A graph is $H$-free if it does not
contain an induced copy of $H$.) Kierstead and Penrice proved it for trees of
radius at most two, but otherwise the conjecture is known only for a few simple
types of forest. More is known if we exclude a complete bipartite subgraph
instead of a clique: R\"odl showed that, for every forest $H$, if $G$ is
$H$-free and does not contain $K_{t,t}$ as a subgraph then it has bounded
chromatic number. In an earlier paper with Sophie Spirkl, we strengthened
R\"odl's result, showing that for every forest $H$, the bound on chromatic
number can be taken to be polynomial in $t$. In this paper, we prove a related
strengthening of the Kierstead-Penrice theorem, showing that for every tree $H$
of radius two and every integer $d\ge 2$, if $G$ is $H$-free and does not
contain as a subgraph the complete $d$-partite graph with parts of cardinality
$t$, then its chromatic number is at most polynomial in $t$.
