  Tree structures have been shown to provide an efficient framework for
propagating beliefs [Pearl,1986]. This paper studies the problem of finding an
optimal approximating tree. The star decomposition scheme for sets of three
binary variables [Lazarsfeld,1966; Pearl,1986] is shown to enhance the class of
probability distributions that can support tree structures; such structures are
called tree-decomposable structures. The logarithm scoring rule is found to be
an appropriate optimality criterion to evaluate different tree-decomposable
structures. Characteristics of such structures closest to the actual belief
network are identified using the logarithm rule, and greedy and exact
techniques are developed to find the optimal approximation.
