We propose the following conjecture: For every fixed α ϵ [0; 1/2], each graph of minimum degree at least (1+α) k/2 and maximum degree at least 2(1−α)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, which disproves a conjecture from [17].