To convey meaning, human language relies on hierarchically organized, long-range relationships spanning words, phrases, sentences, and discourse. The strength of the relationships between sequentially ordered elements of language (e.g., phonemes, characters, words) decays following a power law as a function of sequential distance. To understand the origins of these relationships, we examined long-range statistical structure in the speech of human children at multiple developmental time points, along with non-linguistic behaviors in humans and phylogenetically distant species. Here we show that adult-like power-law statistical dependencies precede the production of hierarchically-organized linguistic structures, and thus cannot be driven solely by these structures. Moreover, we show that similar long-range relationships occur in diverse non-linguistic behaviors across species. We propose that the hierarchical organization of human language evolved to exploit pre-existing long-range structure present in much larger classes of non-linguistic behavior, and that the cognitive capacity to model long-range hierarchical relationships preceded language evolution. We call this the Statistical Scaffolding Hypothesis for language evolution.