E=I+TE=I+T: The internal extent formula for compacted tries

It is well known (Knuth, 1997 [5, pp. 399–400]) that in a binary tree the external path length minus the internal path length is exactly 2n−22n−2, where n is the number of external nodes. We show that a generalization of the formula holds for compacted tries, replacing the role of paths with the notion of extent  , and the value 2n−22n−2 with the trie measure, an estimation of the number of bits that are necessary to describe the trie.

Research highlights► There is a formula relating internal and external path length of a binary tree. ► We generalize the formula so that it applies to compacted binary tries. ► We provide a further generalization that applies to any compacted trie.