The Distance Between the Adjacency Spectral Center and the Characteristic Set of a Tree

Let $\mathcal S(T)$ be the adjacency spectral center of a tree $T$, and let $\mathcal C(T)$ be its characteristic set. We determine the largest possible separation $d(T) := \operatorname{dist}_T(\mathcal{S}(T), \mathcal{C}(T))$ among trees of every order $n\ge3$. Writing $Δ_n:=\max_{|V(T)|=n}d(T)$,