Abstract
Identity trees with bounded maximum degree play a fundamental role in applications-oriented problems, especially when the trees are classified by their diameters. This paper offers results related to enumeration of such tree classes obtained by extending the methods of Gordon and Kennedy [The counting and coding of trees of fixed diameter, SIAM J. Appl. Math. 28 376–398 (1975)]. We set our results into the context of other enumerative work on identity trees. We derive formulae for the numbers of identity trees of various types, with fixed diameter and maximum degree. This then leads to asymptotic formulae (for large diameter). By combining these with formulae derived by Gordon and Kennedy [loc. cit.] we obtain the asymptotic fractions of identity trees among trees in various classes. These fractions are juxtaposed with asymptotic results that have appeared elsewhere. Our final section derives algorithms for integer coding and decoding identity trees in a way that is highly convenient for computer applications.
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