  We propose the following conjecture: For every fixed $\alpha\in [0,\frac
13)$, each graph of minimum degree at least $(1+\alpha)\frac k2$ and maximum
degree at least $2(1-\alpha)k$ contains each tree with $k$ edges as a subgraph.
Our main result is an approximate version of the conjecture for bounded degree
trees and large dense host graphs. We also show that our conjecture is
asymptotically best possible. The proof of the approximate result relies on a
second result, which we believe to be interesting on its own. Namely, we can
embed any bounded degree tree into host graphs of minimum/maximum degree
asymptotically exceeding $\frac k2$ and $\frac 43k$, respectively, as long as
the host graph avoids a specific structure.
