On the number of trees in a random forest

The analytic methods of Polya, as reported in [l, 61 are used to determine the asymptotic behavior of the expected number of (unlabeled) trees in a random forest of orderp. Our results can be expressed in terms of q = .338321856899208.. ., the radius of convergence of t(x) which is the ordinary generating function for trees. We have found that the expected number of trees in a random forest approaches 1 + xFz1 t(~~) = 1.755510... and the form of this result is the same for other species of trees. The problem of estimating the number of trees in a large, random labeled forest was treated in Moon’s book Counting Labeled Trees [3, p. 291. It was found that the average number of labeled trees in all labeled forests ofp points approaches 3/2 as a limit as p increases. We have investigated the same question for unlabeled trees and have found that in this case the average number of trees also approaches a constant, namely 1.755510**. This average an be expressed in terms of the ordinary generating function t(x) for trees and its radius of convergence 7. We use the notation and terminology of the book GraphicaZ Enumeration [I] and the analytic methods of Polya as reported in [ 1, 61. Let F,, be the number of forests of order