The Frame of Nuclei of an Alexandroff Space

Let $\mathcal{O}S$ be the frame of open sets of a topological space $S$, and let $N(\mathcal{O}S)$ be the frame of nuclei of $\mathcal{O}S$. For an Alexandroff space $S$, we prove that $N(\mathcal{O}S)$ is spatial iff the infinite binary tree $\mathscr T_2$ does not embed isomorphically into $(S, \le)$, where $\le$ is the specialization preorder of $S$.