Changes in synaptic connectivity patterns through the formation and elimination of dendritic spines may contribute to structural plasticity in the brain. We characterize this contribution quantitatively by estimating the number of different synaptic connectivity patterns attainable without major arbor remodeling. This number depends on the ratio of the synapses on a dendrite to the axons that pass within a spine length of that dendrite. We call this ratio the filling fraction and calculate it from geometrical analysis and anatomical data. The filling fraction is 0.26 in mouse neocortex, 0.22-0.34 in rat hippocampus. In the macaque visual cortex, the filling fraction increases by a factor of 1.6-1.8 from area V1 to areas V2, V4, and 7a. Since the filling fraction is much smaller than 1, spine remodeling can make a large contribution to structural plasticity.