The center function on trees

When $(X, d)$ is a finite metric space and $\pi = (x_1 , \ldots, x_k ) \in X^k$, a central element for $\pi$ is an element $x$ of $X$ for which max$\{ d(x, x_i ): i = 1 ,\ldots ,k\}$ is minimum. The function that returns the set of all central elements for any tuple $\pi$ is called the center function on $X$. In this note, the center function on finite trees is characterized.