Centered tree


tree.gif|right|frame|On the left a centered tree, on the right a bicentered one.
The numbers show each node's eccentricity. ]
In the mathematical subfield of graph theory, a centered tree is a tree with only one center, and a bicentered tree is a tree with two centers.
Given a graph, the eccentricity of a vertex is defined as the greatest distance from to any other vertex. A center of a graph is a vertex with minimal eccentricity. A graph can have an arbitrary number of centers. However, has proved that for trees, there are only two possibilities:
  1. The tree has precisely one center.
  2. The tree has precisely two centers. In this case, the two centers are adjacent.
A proof of this fact is given, for example, by Harary.