A well-known theorem about infinite trees that if a tree only grows a finite amount at each node, it must have an infinite branch. This is known as KÃ¶nig’s lemma. Obviously, an uncountable tree that only grows a countable amount at each node must have an uncountable branch, right? Amazingly, the answer is no. Counterexamples, now known as Aronszajn trees, were constructed by Nachman Aronszajn. Keith Devlin has more.