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ABSTRACT: Let ae n be the fraction of structures of "size" n which are "connected"; e.g., (a) the fraction of labeled or unlabeled n-vertex graphs having one component, (b) the fraction of partitions of n or of an n-set having a single part or block, or (c) the fraction of n-vertex forests that contain only one tree. Various authors have considered lim ae n , provided it exists. It is convenient to distinguish three cases depending on the nature of the power series for the structures: purely formal, convergent on the circle of convergence, and other. We determine all possible values for the pair (lim inf ae n ; lim sup ae n ) in these cases. Only in the convergent case can one have 0 ! lim ae n ! 1. We study the existence of lim ae n in this case. AMS-MOS Subject Classification (1990): 05A16; Secondary: 05C30, 05C40 1 the electronic journal of combinatorics 7 (2000), #R33 2 1. Introduction Throughout, An will denote the number of structures of size n, Cn will denote the number that are connec...