Twelve Countings with Rooted Plane Trees

The average number of (1) antichains, (2) maximal antichains, (3) chains, (4) infima closed sets, (5) connected sets, (6) independent sets, (7) maximal independent sets, (8) brooms, (9) matchings, (10) maximal matchings, (11) linear extensions, and (12) drawings in (of) a rooted plane tree on n vertices is investigated. Using generating functions we determine the asymptotics and give some explicit formulae and identities. In conclusion we discuss the extremal values of the above quantities and pose some problems. 1 Rooted plane trees A rooted plane tree, a classical enumerative structure, is a quadruple T = (r, V,E, L) such that • (V,E) is a nonempty finite directed tree, as usual V is the vertex set and E is the edge set , • where all edges are directed away from the root r ∈ V , • and L = {({w : vw ∈ E}, <v) : v ∈ V } is a collection of |V | linear orders. We call the elements of the set ch(v) = {w : vw ∈ E} children of v, v is their parent . A leaf is a vertex with no child. Rooted plane trees will be called shortly trees. A tree T is visualized by embedding it in the plane (see Figure 1) so that the root is at the lowest position, all edges are straight segments directed up, and the orders <v coincide with the natural left-right order. By T we denote the collection of all substantially different trees and by Tn the collection of those having n vertices. The aim of the paper is, given a weight w : T → {0, 1, 2, . . .}, to count the total weight w(n) = ∑ T∈Tn w(T ) of trees on n vertices. We consider twelve combinatorial weights w and for the first ten of them we determine the generating function Fw(x) = ∑ T w(T )x|V (T )| = ∑