We consider a distributed detection system with communication constraints, where several nodes are arranged in an arbitrary tree topology, under the assumption of conditionally independent observations. We propose a cyclic design procedure using the minimum expected error probability as a design criterion while adopting a person-by-person methodology. We design each node jointly together with the fusion center, while other nodes are kept fixed, and show that the design of each node using the person-by-person methodology is analogous to the design of a network with two nodes, a network which we refer to as the restricted model. We further show how the parameters in the restricted model for the design of a node in the tree network can be found in a computationally efficient manner. The proposed numerical methodology can be applied for the design of nodes arranged in arbitrary tree topologies with arbitrary channel rates for the links between nodes and for a general M-ary hypothesis testing problem.