Degree conditions for embedding trees

We conjecture that every $n$-vertex graph of minimum degree at least $\frac k2$andmaximumdegreeatleast$2k$containsalltreeswith$k$ 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 $\frac{2}{3}$-conjecture. We prove an approximate version of both conjectures for bounded degree trees and dense host graphs.