Statistical properties of the rooted-tree encoding of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si65.svg" display="inline" id="d1e1724"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:math>

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 investigations and empirical ones with large language models.