Abstract: Abstract A well‐known conjecture of Erdős and Sós states that every graph with average degree exceeding contains every tree with edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding and minimum degree at least contains every tree with edges. As evidence for our conjecture we show (a) for every there is a such that the weakening of the conjecture obtained by replacing the first by holds, and (b) there is a such that the weakening of the conjecture obtained by replacing by holds.