Spanning trees in the square of pseudorandom graphs

We show that for every $\mathrm{\Delta }\in \mathbb{N}$, there exists a constant $C$ such that if $G$ is an $(n,d,\lambda )$-graph with $d\mathrm{/}\lambda \ge C$ and $d$ is large enough, then $G^2$ contains every $n$-vertex tree with maximum degree bounded by $\mathrm{\Delta }$. This answers a question of Krivelevich.