Hereditarily Finite Sets and Identity Trees
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY, Series B 35, 142-155 (1983) Hereditarily Finite Sets and Identity Trees A. MEIR AND J. W. MOON Mathematics Department, University of Alberta, Edmonton, Alberta T6G 2G1, Canada AND J. MYCIELSKI Mathematics Department, University of Colorado, Boulder, Colorado 80309 Communicated by the Editors Received December 21, 1982 Some asymptotic results about the sizes of certain sets of hereditarily finite sets, identity trees, and finite games are proven. 1. INTRODUCTION Harary and Prins [6] introduced a famiiy of rooted trees called identiry trees; these are those trees for which the only automorphism fixing the root is the identity. (See [5] for graph-theoretical terminology not given here.) There is, as we shall see, a natural one-to-one correspondence between the family of identity trees and the family of hereditarily Jinite sets (see [4, pp. 35-36; 3, p. 1681). V arious quantities associated with hereditarily finite sets correspond to quantities associated with identity trees such as (i) the probability that the root has degree k, (ii) the expected degree of the root, and (iii) the expected number of exterior nodes in such a tree. In this paper we consider the asymptotic behaviour of these as well as of some other quan- tities, when the number of vertices in the trees approaches infinity. Before summarizing our results more explicitly, we describe the relation between hereditarily finite sets and identity trees. The family VW of hereditarily finite sets may be defined (see [4, pp.
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