  We conjecture that every $n$-vertex graph of minimum degree at least $\frac
k2$ and maximum degree at least $2k$ contains all trees with $k$ edges as
subgraphs. We prove an approximate version of this conjecture for trees of
bounded degree and dense host graphs.
  Our work also has implications on the Erd\H os--S\'os conjecture and the
$\frac 23$-conjecture. We prove an approximate version of both conjectures for
bounded degree trees and dense host graphs.
