Definition:Center of Tree

This page is about Center in the context of Graph Theory. For other uses, see Center.

Definition

Take a tree, and remove all the nodes whose degree is $1$, along with all their incident edges.

Repeat the process until either:

One node is left

or:

Two nodes are left, joined by a single edge.

If one node is left, it is called the center of the tree.

If two nodes are left, joined by a single edge, this is called the bicenter of the tree.

Central Tree

A tree $T$ is central if and only if it has a center.

Bicentral Tree

A tree $T$ is bicentral if and only if it has a bicenter.

Also see

• Tree has Center or Bicenter, proving every tree has either a center or a bicenter, but not both.

Linguistic Note

The British English spelling of center is centre.

The convention on $\mathsf{Pr} \infty \mathsf{fWiki}$ is to use the American English spelling center, but it is appreciated that there may be lapses.