Phoneme frequency distributions exhibit robust statistical regularities across languages, including exponential-tailed rank-frequency patterns and a negative relationship between phonemic inventory size and the relative entropy of the distribution. The origin of these patterns remains largely unexplained. In this paper, we investigate whether they can arise as consequences of the historical processes that shape phonological systems. We introduce a stochastic model of phonological change and simulate the diachronic evolution of phoneme inventories. A naïve version of the model reproduces the general shape of phoneme rank-frequency distributions but fails to capture other empirical properties. Extending the model with two additional assumptions –an effect related to frequency and a stabilising tendency toward a preferred inventory size– yields simulations that match both the observed distributions and the negative relationship between inventory size and relative entropy. These results suggest that some statistical regularities of phonological systems may arise as a result of diachronic sound change instead of –or in addition to– explicit optimisation or compensatory mechanisms.

In many fields, such as language acquisition, neuropsychology of language, the study of aging, and historical linguistics, corpora are used for estimating the diversity of grammatical structures that are produced during a period by an individual, community, or type of speakers. In these cases, treebanks are taken as representative samples of the syntactic structures that might be encountered. Generalizing the potential syntactic diversity from the structures documented in a small corpus requires careful extrapolation whose accuracy is constrained by the limited size of representative sub-corpora. In this article, I demonstrate—both theoretically and empirically—that a grammar’s derivational entropy and the mean length of the utterances (MLU) it generates are fundamentally linked, giving rise to a new measure, the derivational entropy rate. The mean length of utterances becomes the most practical index of syntactic complexity; I demonstrate that MLU is not a mere proxy, but a fundamental measure of syntactic diversity. In combination with the new derivational entropy rate measure, it provides a theory-free assessment of grammatical complexity. The derivational entropy rate indexes the rate at which different grammatical annotation frameworks determine the grammatical complexity of treebanks. I evaluate the Smoothed Induced Treebank Entropy (SITE) as a tool for estimating these measures accurately, even from very small treebanks. I conclude by discussing important implications of these results for both NLP and human language processing.