The celebrated result of Komlós, Sárközy, and Szemerédi states that for any $\varepsilon>0$, there exists $0<c<1$, such that for all sufficiently large $n$, every $n$-vertex graph $G$ with $δ(G)\geq(1/2+\varepsilon)n$ contains every $n$-vertex tree with maximum degree at most $cn/\log n$. This is best possible up to the value of $c$. In this paper, we extend this result to trees with higher maximum degrees, and prove that for $Δ\gg n/\log n$, roughly speaking, $δ(G)\geq n-n^{1-(1+o(1))Δ/n}$ is the asymptotically optimal minimum degree condition which guarantees that $G$ contains every $n$-vertex spanning tree with maximum degree at most $Δ$. We also prove the corresponding statements in the random graph setting.