Seminar: Generalized plane trees
Speaker: Isaac Owino Okoth, Maseno University, Kenya
Abstract: Many classical families of trees in mathematics, such as plane trees and noncrossing trees, can be described by how they are embedded in the plane. A key notion in this area is the butterfly, introduced by Flajolet and Noy, which consists of a pair of noncrossing trees sharing a common root, with each tree called a wing. Building on this concept, we introduce a new class of trees that generalizes both plane trees and noncrossing trees by placing all vertices on the boundary of a circle and drawing edges as straight-line segments that do not cross inside the circle, with the additional condition that a butterfly rooted at any vertex has at most d wings. We call these structures d-dimensional plane trees and enumerate them with respect to parameters including root degree, numbers of vertices of a given type, total number of vertices, leaves, endpoints, boundary edges, descents, forests, and degree sequences, thereby unifying and extending known results for plane trees and noncrossing trees.
Keywords: Trees, Non-Crossing Trees, Parameters on Trees