We prove that, if m is sufficiently large, every graph on m + 1 vertices that has a universal vertex and minimum degree at least left perpendicular2m/3right perpendicular 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.