Statistical Properties of the Rooted-Tree Encoding of $\mathbb{N}$
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
Keywords
Cite
@article{arxiv.2512.01436,
title = {Statistical Properties of the Rooted-Tree Encoding of $\mathbb{N}$},
author = {Pierluigi Contucci and Claudio Giberti and Godwin Osabutey and Cecilia Vernia},
journal= {arXiv preprint arXiv:2512.01436},
year = {2025}
}