A well-known conjecture of Erd\H{o}s and S\'os states that every graph with
average degree exceeding m−1 contains every tree with m edges as a
subgraph. We propose a variant of this conjecture, which states that every
graph of maximum degree exceeding m and minimum degree at least ⌊32m⌋ contains every tree with m edges.
As evidence for our conjecture we show (i) for every m there is a g(m)
such that the weakening of the conjecture obtained by replacing m by g(m)
holds, and (ii) there is a γ>0 such that the weakening of the conjecture
obtained by replacing ⌊32m⌋ by (1−γ)m holds