Structural Complexity of the Rooted-Tree Factorization of Integers

We encode natural numbers as planar rooted trees via iterated Euclidean decomposition, yielding a fully deterministic symbolic text of Dyck words governed by intrinsic syntactic rules. Without introducing probabilistic assumptions or generative models, we quantify how information, redundancy, and correlations are distributed within this corpus. Both the dictionary of distinct Dyck words and the entropy grow sublinearly, while the rank–frequency curve exhibits a stable parabolic fractal form indicative of hierarchical self-similarity rather than Zipfian scaling. Correlation measurements based on anomalous diffusion reveal a transition from near-diffusive to quasi-ballistic regimes, demonstrating persistent structural organization across scales. Together, these findings show that the sequence of natural numbers, when expressed through its tree-based Dyck-word representation, forms a highly non-random, statistically structured text in which large-scale order and predictability emerge directly from arithmetic determinism.