Spanning trees with large maximum degrees

The celebrated result of Komlós, Sárközy, and Szemerédi states that for any $\epsilon >0$, there exists $0<c<1$, such that for all sufficiently large $n$, every $n$-vertex graph $G$ with $\delta (G)\ge (1\mathrm{/}2+\epsilon )n$ contains every $n$-vertex tree with maximum degree at most $cn\mathrm{/}\log n$. 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 $\mathrm{\Delta }\gg n\mathrm{/}\log n$, roughly speaking, $\delta (G)\ge n-n^{1-(1+o(1))\mathrm{\Delta }\mathrm{/}n}$ is the asymptotically optimal minimum degree condition which guarantees that $G$ contains every $n$-vertex spanning tree with maximum degree at most $\mathrm{\Delta }$. We also prove the corresponding statements in the random graph setting.