In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, 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.