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Adaptive avian radiations associated with the diversification of bird beaks into a multitude of forms enabling different functions are exemplified by Darwin's finches and Hawaiian honeycreepers. To elucidate the nature of these radiations, we quantified beak shape and skull shape using a variety of geometric measures that allowed us to collapse the variability of beak shape into a minimal set of geometric parameters. Furthermore, we find that just two measures of beak shape—the ratio of the width to length and the normalized sharpening rate (increase in the transverse beak curvature near the tip relative to that at the base of the beak)—are strongly correlated with diet. Finally, by considering how transverse sections to the beak centreline evolve with distance from the tip, we show that a simple geometry-driven growth law termed 'modified mean curvature flow' captures the beak shapes of Darwin's finches and Hawaiian honeycreepers. A surprising consequence of the simple growth law is that beak shapes that are not allowed based on the developmental programme of the beak are also not observed in nature, suggesting a link between evolutionary morphology and development in terms of growth-driven developmental constraints. [ABSTRACT FROM AUTHOR]

Although an increasing number of studies are combining skeletal and neural morphology data in a phylogenetic context, most studies do not include extinct taxa due to the rarity of preserved endocasts. The early Eocene avifauna of the Fur Formation of Denmark presents an excellent opportunity for further study of extinct osteological and endocranial morphology as fossils are often exceptionally preserved in three dimensions. Here, we use X-ray computed tomography to present additional material of the previously described taxon Scandiavis mikkelseni and reassess its phylogenetic placement using a previously published dataset. The new specimen provides novel insights into the osteological morphology and brain anatomy of Scandiavis. The virtual endocast exhibits a morphology comparable to that of modern avian species. Endocranial evaluation shows that it was remarkably similar to that of certain extant Charadriiformes, yet also possessed a novel combination of traits. This may mean that traits previously proposed to be the result of shifts in ecology later in the evolutionary history of Charadriiformes may instead show a more complex distribution in stem Charadriiformes and/or Gruiformes depending on the interrelationships of these important clades. Evaluation of skeletal and endocranial character state changes within a previously published phylogeny confirms both S. mikkelseni and a putative extinct charadriiform, Nahmavis grandei, as charadriiform. Results bolster the likelihood that both taxa are critical fossils for divergence dating and highlight a biogeographic pattern similar to that of Gruiformes. [ABSTRACT FROM AUTHOR]