This paper presents a framework for studying the centralized optimal multi-agent coordination problem under tree formation constraints. The geodesic equations characterizing the optimal coordinated motions are derived in a suitably chosen coordinate system for general tree formation constraints. The solutions to these equations, however, may fail to be optimal once extended beyond certain points called the conjugate points due to the failure of the second-order optimality condition. For the particular class of star formations, two methods for computing the conjugate points along a natural candidate solution are introduced. Using these methods, we derive analytically the conjugate points, as well as the better solutions once the candidate solution is extended beyond its first conjugate point. The optimal centralized coordinated motions derived in this paper will yield a performance lower bound for those generated by decentralized algorithms.