The length and spacing of river tributaries.

River networks are composed of a mainstem and tributaries. These tributaries dissect landscapes, regulate water and habitat availability, and transport sediment and nutrients. Despite the importance of tributaries, we currently lack theory and data describing whether and how tributary length and spacing varies within watersheds, thereby limiting our ability to accurately describe river network geometry. We address this knowledge gap by analyzing 4,696 tributaries across six landscapes with varying climate, tectonic setting, and lithology. Our results show that both tributary length and spacing systematically increase with downstream distance along the mainstem river, following a power-law scaling. This power-law scaling can be modulated by basin shape, with tributaries becoming shorter and, in some cases, more closely spaced as basin elongate. Furthermore, the power-law scaling may break down in cases where river networks have been disturbed by pervasive faulting, raising the possibility that the scaling we observe is not unique to all branching networks, and instead may be universal across undisturbed fluvial networks. These findings can be used to improve predictions of river network geometry and potentially to distinguish fluvial river networks from other branching networks.