In discrete mathematics, a centered tree is a tree

Tree (graph theory)

In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...

 with only one center

Graph center

The center of a graph is the set of all vertices of minimum eccentricity, that is, the set of all vertices A where the greatest distance d to other vertices B is minimal. Equivalently, it is the set of vertices with eccentricity equal to the graph's radius...

, and a bicentered tree is a tree with two centers.

Given a graph, the eccentricity of a vertex v is defined as the greatest distance

Distance (graph theory)

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path connecting them. This is also known as the geodesic distance...

from v to any other vertex. A center (also: centroid)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 (centered trees).
2. The tree has precisely two centers (bicentered trees). In this case, the two centers are adjacent.

A proof of this fact is given, for example, by Knuth.