In this paper and a companion paper, we prove that, if m $m$ is sufficiently large, every graph on m + 1 $m+1$ vertices that has a universal vertex and minimum degree at least L 2 m 3 RIGHT FLOOR $\lfloor \phantom{\rule[-0.5em]{}{0ex}}\frac{2m}{3}\rfloor $ contains each tree T $T$ with m $m$ edges as a subgraph. Our result confirms, for large m $m$, 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.