Center of a tree ($C(T)=\arg {\min }_{v}e(v)$)

The center of a finite unweighted tree is the set of vertices minimizing graph eccentricityIt 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 $a,b$: $D\le d(a,v)+d(v,b)$, 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.