On graphs which contain all small trees

We investigate those graphs G_n with the property that any tree on n vertices occurs as subgraph of G_n. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by $\text{1}\text{2}\text{n}\text{log}\text{n}$ and $\text{n}{}^{\mathrm{<mtext>1+</mtext><mtext>1</mtext><mtext>log\; log</mtext><mtext>\; n</mtext>}}$, respectively.