Spanning trees in graphs of high minimum degree with a universal vertex I: An asymptotic result

In this paper and a companion paper, we prove that, if is sufficiently large, every graph on vertices that has a universal vertex and minimum degree at least contains each tree with edges as a subgraph. Our result confirms, for large , an important special case of a recent conjecture by Havet, Reed, Stein and Wood. The present paper already contains an approximate version of the result.