Twenty-step algorithm for determining the asymptotic number of trees of various speces

The technique for finding the asymptotic number of unlabeled trees of various sorts was developed by Polya (1937) and perfected by Otter (1948). Modern presentations are available in the book of Harary and Palmer (1973; Chapter 9), and in the paper of Bender (to appear). An exposition of the basic method is here developed in the form of a 20 step algorithm, which should facilitate the finding of asymptotic formulas for different kinds of trees. These 20 steps are presented in Section 2, and methods of justifying the steps are supplied in Section 3. In Sections 4, 5 and 6, the algorithm is applied to finding asymptotic values for the number of identity trees, homeomorphically irreducible trees, and a class of blocks with tree-like properties. The first two of these species were enumerated by Harary and Prins (1959) and the third is easily done. However, no asymptotic analyses have been given previously. For the purpose of the discussion in Sections 2 and 3, a hypothetical class lof trees is posed, of which there are Sn planted trees on n + 1 points (including [the root which is an endpoint; hence there are n lines) and sn unrooted trees on In points. We let S(x) and s(x) be the ordinary generating functions