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The center of a graph is the set of vertices with minimum graph eccentricity. For a finite connected graph $G$, putHere $d_G$ is shortest-path distance and $\mathrm{rad}G={\min }_{v}e(v)$ is the graph radius. The center solves a facility-placement problem in which the cost is the distance to the farthest vertex.

In a five-vertex path graph, eccentricities are $4,3,2,3,4$, so only the middle vertex is central. In a four-vertex path, the two middle vertices both have eccentricity two. The center of a tree is always a single vertex or a pair of adjacent vertices: repeatedly deleting all leaves exposes its middle. For a general graph the center need not have this form; every vertex of a cycle graph is central.

Minimizing the sum of distances defines a different objective. For example, let $a,b,c,d$ form a path, and attach three additional leaves to $a$. The center is $\{b\}$, but the sum of distances is smallest at $a$. The farthest-distance criterion and the average-distance criterion can therefore select different locations.

For finite nonempty trees, the dissymmetry theorem for trees gives an isomorphism of combinatorial classeswhere the superscripts mark a vertex, an undirected edge, or a directed edge. The tree on each side retains the same labels or other decorations; the identity is stronger than a count of labeled trees.

Repeatedly remove all leaves until one vertex or one edge remains. This is the invariant center of a tree. If a marked vertex or edge is the center, forget the mark and obtain the unrooted tree. Otherwise use the unique path toward the center: a marked vertex determines the first edge oriented toward the center, while a marked edge is oriented away from the center. These assignments give all directed-edge marks exactly once and can be reversed from the center and the orientation. This supplies the required bijection.

For a fixed tree with $n$ distinctly labeled vertices, the counts reduce to $1+2(n-1)=n+(n-1)$. For unlabeled trees, symmetry makes the bijection essential: one cannot simply divide a rooted count by $n$.