Spanning trees in graphs of high minimum degree with a universal vertex II: A tight result

We prove that, ifm $m$ is sufficiently large, every graph onm + 1 $m+1$ vertices that has a universal vertex and minimum degree at least⌊2 m 3 ⌋$\lfloor \phantom{\rule[-0.5em]{}{0ex}}\frac{2m}{3}\rfloor $ contains each treeT $T$ withm $m$ edges as a subgraph. Our result confirms, for largem $m$ , 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.