We examine in detail the causes of the structural transitions that occur for those small Lennard-Jones clusters that have a nonicosahedral global minima. Based on the principles learned from these examples, we develop a method to construct structural phase diagrams that show in a coarse-grained manner how the equilibrium structure of large clusters depends on both size and temperature. The method can be augmented to account for anharmonicity and quantum effects. Our results illustrate that the vibrational entropy can play a crucial role in determining the equilibrium structure of a cluster.