Tree Dimension and the Sauer-Shelah Dichotomy

TL;DR

This paper introduces tree dimension concepts to analyze the complexity of leaf sets in binary and higher-arity trees, providing new bounds and insights into classical combinatorial lemmas like Sauer-Shelah.

Contribution

It defines tree dimension and leveled variants, establishes tight bounds on leaf set sizes, and unifies classical results through a new combinatorial framework.

Findings

Provides tight upper bounds on leaf set sizes using leveled tree dimension

Unifies Sauer-Shelah Lemma and Bhaskar's Littlestone dimension result

Classifies maximal leaf sets by tree dimension and extends to higher-arity trees

Abstract

We introduce tree dimension and its leveled variant in order to measure the complexity of leaf sets in binary trees. We then provide a tight upper bound on the size of such sets using leveled tree dimension. This, in turn, implies both the famous Sauer-Shelah Lemma for VC dimension and Bhaskar's version for Littlestone dimension, giving clearer insight into why these results place the exact same upper bound on their respective shatter functions. We also classify the isomorphism types of maximal leaf sets by tree dimension. Finally, we generalize this analysis to higher-arity trees.