Comparative studies have established differences between the electrophysiology and anatomy of human and rodent cortical circuits. A consistent finding is that human neuronal morphologies display more elaborate neurite shapes than those of rodents, a feature that cannot be accounted for merely by their larger size according to recent findings. Here, we study the impact of these neurite shapes on the structure of synaptic connectivity in their local microcircuitry. Our approach is based on the idea that axonal and dendritic geometries constrain the locations of afferent and efferent synaptic contacts (potential connectivity). Although the mechanisms by which potential connectivity translates into actual synaptic connectivity are manifold and complex, the potential connectivity is nevertheless highly informative for the final structure of a biological connectome. We found that connectomes predicted from human reconstructed morphologies have higher complexity according to several measures that have been demonstrated to be functionally relevant. Going beyond a simple comparison, we demonstrate mechanistically how the shapes of neuron morphologies give rise to non-random and clustered structures observed in experimentally measured connectomes, and how the specific shapes of human neurons strengthen the process. Finally, we conceptually examine how synapse formation processes may interact with potential connectivity, showing that a process compatible with Hebbian plasticity leads to the highest complexity and best match experimentally observed patterns.