= Center of a tree
{title2=$C(T)=\arg\min_ve(v)$}
The center of a finite unweighted <tree> is the set of vertices minimizing <graph eccentricity>
$$
e(v)=\max_{u\in V(T)}d_T(v,u).
$$
It consists of either one vertex or two adjacent vertices. If a longest path has $D$ edges, its middle vertex or middle pair is the center, and the minimum eccentricity is $\lceil D/2\rceil$.
The lower bound follows from the path endpoints $a,b$: $D\leq d(a,v)+d(v,b)$, so one endpoint is at least $D/2$ away from $v$. A middle vertex attains the bound; a branch extending farther from it would produce a path longer than the chosen diameter path.
A practical construction repeatedly deletes all leaves simultaneously until only one or two vertices remain. Each pruning removes the outermost distance layer without changing the eventual middle. For a star the center is its hub; for a four-vertex path it is the two internal vertices. Weighted or infinite trees require their own metric and existence conventions.
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