Snow avalanches often form levees and en-echelon shear planes in the run-out zone. We describe the formation of these depositional structures using a simple model that accounts for the role of granular fluctuations in avalanche motion. A mathematical feature of this model is the existence of a bifurcation saddle point, describing how granular fluctuations control the avalanche velocity in the run-out zone. The saddle point discriminates between a flowing and stopping regime and defines the physical boundary between the flow and non-flow regions of the avalanche, i.e. the location of shear planes in the avalanche deposits. The formation of a shear plane depends on the interplay between terrain slope and avalanche mass flux, which varies from avalanche head to tail. Levees can form immediately at the avalanche front or, for steep slopes and low mass fluxes, at the avalanche tail. At ravine and gully shoulders the mass flux is restricted, thus initiating levee formation. We find that the levee lines are parallel to the flow direction when the mass flux is constant; en-echelon shear lines occur when the mass flux is decreasing. We test the model using several case studies where we have accurate laser scans of avalanche deposits. Our results suggest that avalanche flow parameters can be determined from simple levee measurements or, conversely, formation of levees and flow fingers can be predicted once the parameters governing the granular fluctuations are known.