Various methods have been given for establishing the existence of the free inverse semigroup FIA on a set A, and for constructing it explicitly (see, for example, [2], [5], [7], [9], [10], [11]). In this paper we outline a graph-theoretic technique for representing the elements of FIA. This depends on the notion, introduced here, of a word-tree on A. With the aid of this technique various properties of FIA are easily deduced: some of these are stated below.