Common developmental origins of beak shapes and evolution in theropods.

Vertebrate beaks show a remarkable diversity of forms, epitomized by birds and non-avian theropods. Few studies have investigated how underlying developmental processes influence beak shape. The power cascade is a model of growth describing the log-log linear relationship of the beak radius with distance from the tip. We measured beak and toothed snout shapes in 127 species across 120 families of extant birds and extinct non-avian theropods and found that 95% followed the power cascade model. Ancestral state estimation suggests that the power cascade constitutes a fundamental growth pattern of the theropod rostrum, and perhaps all vertebrate rostra. The morphospace defined by the power cascade shows how bird beak shape explores the geometries associated with ecological specializations while adhering to the growth model. We show that the power cascade influences the macroevolutionary exploration of rostrum morphospace, enabling extant birds to inhabit all components of Earth's biosphere.