IfG andH are graphs, let us writeG→(H)2 ifG contains a monochromatic copy ofH in any 2-colouring of the edges ofG. Thesize-Ramsey numberre(H) of a graphH is the smallest possible number of edges a graphG may have ifG→(H)2. SupposeT is a tree of order |T|≥2, and lett0,t1 be the cardinalities of the vertex classes ofT as a bipartite graph, and let Δ(T) be the maximal degree ofT. Moreover, let Δ0, Δ1 be the maxima of the degrees of the vertices in the respective vertex classes, and letβ(T)=T0Δ0+t1Δ1. Beck [7] proved thatβ(T)/4≤re(T)=O{β(T)(log|T|)12}, improving on a previous result of his [6] stating thatre(T)≤Δ(T)|T|(log|T|)12. In [6], Beck conjectures thatre(T)=O{Δ(T)|T|}, and in [7] he puts forward the stronger conjecture thatre(T)=O{β(T)}. Here, we prove the first of these conjectures, and come quite close to proving the second by showing thatre(T)=O{β(T)logΔ(T)}.