For every $k\ge 2$ and $Δ$, we prove that there exists a constant $C_{Δ,k}$ such that the following holds. For every graph $H$ with $χ(H)=k$ and every tree with at least $C_{Δ,k}|H|$ vertices and maximum degree at most $Δ$, the Ramsey number $R(T,H)$ is $(k-1)(|T|-1)+σ(H)$, where $σ(H)$ is the size of a smallest colour class across all proper $k$-colourings of $H$. This is tight up to the value of $C_{Δ,k}$, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.