  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 conjecture by Havet,
Reed, Stein, and Wood. The present paper builds on the results of a companion
paper in which we proved the statement for all trees having a vertex that is
adjacent to many leaves.
