  Let H be a tree. It was proved by Rodl that graphs that do not contain H as
an induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$
as a subgraph, have bounded chromatic number. Kierstead and Penrice
strengthened this, showing that such graphs have bounded degeneracy. Here we
give a further strengthening, proving that for every tree H, the degeneracy is
at most polynomial in t. This answers a question of Bonamy, Pilipczuk,
Rzazewski, Thomasse and Walczak.
