We denote $\displaystyle C(T)$ and $\displaystyle D(T)$ to be the center set and centroid set respectively of a tree $\displaystyle T$. For each $\displaystyle n \geq 2$, construct a tree $\displaystyle T$ in which $\displaystyle \text{dist}_T(C(T), D(T)) \geq n$. I don't see how to construct these trees.