We provide a unified framework for the treatment of special integrable systems which we propose to call 'generalized mean-field systems'. Thereby previous results on integrable classical and quantum systems are generalized. Following Ballesteros and Ragnisco, the framework consists of a unital algebra with brackets, a Casimir element and a coproduct which can be lifted to higher tensor products. The coupling scheme of the iterated tensor product is encoded in a binary tree. The theory is exemplified by the case of a spin octahedron. The relation to other generalizations of the coalgebra approach is discussed.