# Definition:Center of Tree

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*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:

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.