Year: 2023
Author: Lin Yang, Shengliang Yang
Journal of Mathematical Study, Vol. 56 (2023), Iss. 1 : pp. 1–17
Abstract
In this paper, we consider the class of ordered trees and its two subclasses, bushes and planted trees, which consist of the ordered trees with root degree at least $2$ and with root degree $1$ respectively. In these three classes, we study the number of trees of size $n$ with $k$ protected (resp. unprotected) branches, and the total number of branches (resp. protected branches, unprotected branches) among all trees of size $n$. The explicit formulas as well as the generating functions are obtained. Furthermore, we find that, in each class, as $n$ goes to infinity, the proportion of protected branches among all branches in all trees of size $n$ approaches $ 1/3$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v56n1.23.01
Journal of Mathematical Study, Vol. 56 (2023), Iss. 1 : pp. 1–17
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 17
Keywords: ordered tree bush planted tree protected branch unprotected branch Catalan number generating function.