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

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.

Linguistic Note
Note that the diacritics on the ö in the König of König's Tree Lemma and on the ő of do not match. Unfortunately, this is how it is.