THE NUMBER OF PLANAR TREES
David W. Walkup · 1969
Abstract : By a planar tree is meant a realization of a tree in the plane, and by an isomorphism between two planar trees is meant a mapping which is not only an isomorphism in the usual sense of trees but which also preserves the clockwise cyclic order of edges about each node. Explicit formulae are given for each of the following: (1) the number of nonisomorphic unrooted planar trees with n edges, (2) the number of nonisomorphic rooted planar trees with n edges, and (3) the number of nonisomorphic rooted planar trees with n edges such that the root is incident on exactly k edges, of which one is distinguished.