Several researchers (Beeri, Bernstein, Chiu, Fagin, Goodman, Maier, Mendelzon, Ullman, and Yannakakis) have introduced a special class of database schemes, called acyclic or tree schemes. Beeri et al. have shown that an acyclic join dependency, naturally defined by an acyclic database scheme, has several desirable properties, and that an acyclic join dependency is equivalent to a conflict-free set of multivalued dependencies. However, since their results are confined to multivalued and join dependencies, it is not clear whether we can handle functional dependencies independently of other dependencies. In the present paper we define an extension of a conflict-free set, called an extended conflict-free set , including multivalued dependencies and functional dependencies, and show the following two properties of an extended conflict-free set: There are three equivalent definitions of an extended conflict-free set. One of them is defined as a set including an acyclic joint dependency and a set of functional dependencies such that the left and right sides of each functional dependency are included in one of the attribute sets that construct the acyclic join dependency. For a relation scheme with an extended conflict-free set, there is a decomposition into third normal form with a lossless join and preservation of dependencies.