Definitions and notation for trees.
. If , and , . If , the concatenation of and is defined by . If , , then is an initial segment of and is an extension of (written ), if , for some .
A tree on a set is a subset closed under initial segments. The body of , . An element of is called terminal if and for all .
Let is a tree on . is finite splitting if for every there only finitely many such that . is pruned if every has a proper extension .
Fact. If is finite splitting, for every , the number of strings in of length is finite.
Assume that for some , the number of strings in of length is infinite. Then, there must be a minimal number , which satisfies the property that infinitely many of the strings differ at position , but agree at all previous positions. Then is not finite splitting.