On Tue, 29 Apr 2003, N. J. A. Sloane wrote:

Isn't that the subject of the following well-known article?

%D A003018 R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis. Amer. Math. Monthly 80 (1973), 868-876.

(there are numerous references to it in the OEIS)

Yes, but none linked to A000081 (Rooted Trees). The observation that the number of distinct functions obtained by inserting parentheses in x^x^x^...^x where there are n x's is equal to the number of rooted trees with n nodes apparently appeared first in an earlier paper (referenced by Guy and Selfridge): F. G\"{o}bel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103 May I suggest that these two references be added to the references for A000081 (rooted trees) and that this interpretation of sequence A000081 be added to the comments with a link to A003018. --Edwin Clark