-trees and Applications

Mahendra Jani and Melkamu Zeleke  

Department of Mathematics, William Paterson University, Wayne, NJ 07470, U.S.A.

JaniM@wpunj.edu  ZelekeM@wpunj.edu

Abstract.      Full Text PDF

A K-tree is constructed from a single distinguished k-cycle by repeatedly gluing other k-cycles to existing ones along an edge. If K is a nonempty subset of , then a K-tree is obtained similarly using k-cycles with . In this talk, we enumerate ordered K-trees, and show that the ratio of terminal edges to total number of edges in k-trees is . Then, using K-trees as models we enumerate planted plane cacti and generalize the Tennis Ball Problem. A summation formula for all possible arrangements of Tennis Balls is also obtained using K-trees and lattice paths.

Finally, we use -trees to obtain generating function identities involving generalizations of Catalan, Central Binomial, and Fine Numbers. We give examples to show possible applications of these identities and show that the ratio of generalized Fine numbers to Catalan numbers is asymptotic to .