The butterfly decomposition of plane trees

Abstract

We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition which turns out to have many applications. From the butterfly decomposition we obtain a oneto-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial coefficients. We also establish a one-to-one correspondence between leaf-colored doubly rooted plane trees and free Schröder paths. The classical Chung-Feller theorem on free Dyck paths and some generalizations and variations with respect to Dyck paths and Schröder paths with flaws turn out to be immediate consequences of the butterfly decomposition and the preorder traversal of plane trees. We obtain two involutions on free Dyck paths and free Schröder paths, leading to two combinatorial identities. We also use the butterfly decomposition to give a combinatorial treatment of the generating function for the number of chains in plane trees due to Klazar. We further study the average size of chains in plane trees with n edges and show that this number asymptotically tends to n+9 6 .

On three and four vicious walkers

“…The second step is to show that 1 þDðtÞ is given by Chen et al (2007) in terms of doubly rooted plane trees and the butterfly decomposition.…”

On the combinatorics of the Pfaff identity

“…A free Dyck path of semilength n is a lattice path from the origin to (2n, 0) consisting of up steps (1, 1) and down steps (1, −1), whereas a Dyck path is a free Dyck path that does not go below the x-axis. Free Dyck paths have been studied in [5] in connection with the enumeration of plane trees. A free Schröder path of semilength n is a lattice path from (0, 0) to (2n, 0) with up steps (1, 1), horizontal steps (2, 0) and down steps (1, −1).…”

“…The weight of a path is the product of the weights of the steps; the weight of a set of paths means the sum of the weights of the paths. Weighted lattices have been used to give combinatorial interpretations of combinatorial identities, see, for example [5,6]. To connect the Simons identity to lattice paths, we use the following rule to assign the weight of a free Schröder path:…”

“…Munarini [20] derived the following generalization of Simons' identity by using Cauchy's integral formula, 5) which can be restated as…”

Bijections for a class of labeled plane trees

“…The enumeration of these and many other families of lattice paths was treated in [1]. For k = 1, our bijection reduces to the classic glove bijection [2,4,18] between plane trees and Dyck paths.…”