König's Tree Lemma

Theorem
Let $T$ be a rooted tree with an infinite number of nodes, each with a finite number of children.

Then $T$ has a branch of infinite length.

Note
This result may not hold if there exists at least one node which has an infinite number of children.

For example, let $T$ be the rooted tree defined as follows:


 * $t_0$ is the root node.


 * For all $n \in \N: n > 0$: $t_n$ is a leaf node which is a child of $t_0$.

Then there is an infinite number of nodes of $T$.

However, each branch of $T$ is of length equal to $1$.

Also see
This is a special case of the trickier to prove König's Lemma, which is a result that applies to all connected infinite graphs whose nodes are all finite in degree.

Also known as
This theorem is also referred to as König Tree Lemma, König's Tree Theorem and König Tree Theorem.