Flexible plants, fungi and sessile animals reconfigure in wind and water to reduce the drag acting upon them. In strong winds and flood waters, for example, leaves roll up into cone shapes that reduce drag compared with rigid objects of similar surface area. Less understood is how a leaf attached to a flexible leaf stalk will roll up stably in an unsteady flow. Previous mathematical and physical models have only considered the case of a flexible sheet attached to a rigid tether in steady flow. In this paper, the dynamics of the flow around the leaf of the wild ginger Hexastylis arifolia and the wild violet Viola papilionacea are described using particle image velocimetry. The flows around the leaves are compared with those of simplified physical and numerical models of flexible sheets attached to both rigid and flexible beams. In the actual leaf, a stable recirculation zone is formed within the wake of the reconfigured cone. In the physical model, a similar recirculation zone is observed within sheets constructed to roll up into cones with both rigid and flexible tethers. Numerical simulations and experiments show that flexible rectangular sheets that reconfigure into U-shapes, however, are less stable when attached to flexible tethers. In these cases, larger forces and oscillations due to strong vortex shedding are measured. These results suggest that the three-dimensional cone structure in addition to flexibility is significant to both the reduction of vortex-induced vibrations and the forces experienced by the leaf.