# Definition:Tree (Graph Theory)

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

## Definition

### Definition 1

A **tree** is a simple connected graph with no circuits.

### Definition 2

A **tree** is a simple connected graph with no cycles.

### Node

The vertices of a tree are called its **nodes**.

### Leaf Node

Let $v$ be a node of a tree $T$.

Then $v$ is a **leaf node** of a $T$ if and only if $v$ is of degree $1$.

If $T$ is a rooted tree, this is equivalent to saying that $v$ has no child nodes.

## Finite or Infinite

### Finite Tree

A tree is **finite** if and only if it contains a finite number of nodes.

### Infinite Tree

A tree is **infinite** if and only if it contains a (countably) infinite number of nodes.

## Also defined as

In some contexts, the term **tree** is used to mean **rooted tree**.

## Also known as

Some sources refer to a **tree** as a **free tree** or an **unrooted tree** to distinguish it specifically from a **rooted tree**.

## Examples

### Arbitrary Example 1

## Also see

- Results about
**trees**can be found**here**.

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- All trees are bipartite, from Graph is Bipartite iff No Odd Cycles (as a
**tree**has no cycles at all).

- No (non-edgeless)
**tree**is Eulerian, as it has no circuits, let alone Eulerian ones.