  We show that for any $\varepsilon>0$ and $\Delta\in\mathbb{N}$, there exists
$\alpha>0$ such that for sufficiently large $n$, every $n$-vertex graph $G$
satisfying that $\delta(G)\geq\varepsilon n$ and $e(X, Y)>0$ for every pair of
disjoint vertex sets $X, Y\subseteq V(G)$ of size $\alpha n$ contains all
spanning trees with maximum degree at most $\Delta$. This strengthens a result
of B\"ottcher et al.
