For every k >= 2 and Delta, we prove that there exists a constant C Delta,k such that the following holds. For every graph H with chi(H) = k and every tree T with at least C Delta,k|H| vertices and maximum degree at most Delta, the Ramsey number R(T, H) is (k - 1)(|T|- 1) + sigma(H), where sigma(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 Delta,k, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.