In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

subfield of

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, a centered tree is a

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only ...

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 Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...

from to any other vertex. A ''center'' of a graph is a vertex with minimal

eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...

. A graph can have an arbitrary number of centers. However, has proved that for trees, there are only two possibilities: # The tree has precisely one center (centered trees). # 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 Harary., Theorem 4.2

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