Statistical Properties of the Rooted-Tree Encoding of $\mathbb{N}$

TL;DR

This paper analyzes the statistical properties of a novel rooted-tree encoding of natural numbers, revealing complex patterns, correlations, and deviations from Zipf's law, with implications for understanding structured sequences.

Contribution

It introduces a recursive tree-based encoding of natural numbers and provides the first detailed statistical analysis of its properties and correlations.

Findings

Dictionary and entropy grow sublinearly

Compression exhibits non-monotonic trends

Rank-frequency curves are parabolic, deviating from Zipf's law

Abstract

We prime-encode the natural numbers via recursive factorisation, iterated to the exponents, generating a corpus of planar rooted trees equivalently represented as Dyck words. This forms a deterministic text endowed with internal rules. Statistical analysis of the corpus reveals that the dictionary and the entropy grow sublinearly, compression shows non-monotonic trend, and the rank-frequency curves assume a stable parabolic form deviating from Zipf's law. Correlation analysis using mean-squared displacement reveals a transition from normal diffusion to superdiffusion in the associated walk. These findings characterise the tree-encoded sequence as a statistically structured text with long-range correlations grounded in its generative arithmetic law, providing an empirical basis for subsequent theoretical and learnability