  In arXiv:1302.4423, Salez proved that every totally real algebraic integer is
the eigenvalue of some tree. We define the "arboreal height" of a totally real
algebraic integer $\lambda$ to be the minimal height of a rooted tree having
$\lambda$ as an eigenvalue. In this paper, we prove several results about
totally real algebraic integers of arboreal height $2$: We show that all real
quadratic integers have arboreal height $\le 2$. We characterize the totally
real cubic integers of arboreal height $2$. Finally, we prove that every
totally irrational real number field is generated (as a ring over $\mathbb{Q}$)
by some $\lambda$ of arboreal height $2$.
