Ramsey numbers of trees

We show that there exists a constant $c>0$ such that every $n$-vertex tree $T$ with $\Delta(T)\le cn$ has Ramsey number $R(T)=\max\{t_1+2t_2,2t_1\}-1$, where $t_1\ge t_2$ are the sizes of the bipartition classes of $T$. This improves an asymptotic result of Haxell, {\L}uczak, and Tingley from 2002, and shows that, though Burr's 1974 conjecture on the Ramsey numbers of trees has long been known to be false for certain `double stars', it is true for trees with up to small linear maximum degree.