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

TL;DR

This paper generalizes a known binary tree property to compacted tries, relating their extent measure to the number of bits needed for description, thus providing a new theoretical insight into trie structure analysis.

Contribution

It introduces a generalized formula linking extent and trie measure for compacted tries, extending classical binary tree path length relations.

Findings

The formula holds for compacted tries, connecting extent and trie measure.

Provides a theoretical basis for analyzing trie complexity.

Enhances understanding of trie structure and description length.

Abstract

It is well known that in a binary tree the external path length minus the internal path length is exactly 2n-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-2 with the trie measure, an estimation of the number of bits that are necessary to describe the trie.