Information Theory of Random Composition Trees

Random composition trees model the hierarchical semantic structures of meaningful texts as a statistical ensemble generated by recursively splitting texts into coherent chunks. A recent study shows that different text types can be described by random composition trees with distinct branching factors, and that the entropy rates of these trees exhibit a similar dependence across text types as the entropy rates of the actual texts. Motivated by this correspondence, we analyze information-theoretic properties of this stochastic splitting model. First, we demonstrate that the entropy of random composition trees grows linearly with the root size, and we derive an analytical expression for the entropy rate as a function of the branching factor. Second, we prove that large trees exhibit typicality: with high probability, they belong to a typical set in which each tree is approximately equally likely. Our analysis reveals that the random composition tree model shares two key information-theoretic properties of language when modeled as a stationary and ergodic stochastic process: linear entropy growth and typicality.