TY - JOUR
T1 - Protected Branches in Ordered Trees
AU - Yang , Lin
AU - Yang , Shengliang
JO - Journal of Mathematical Study
VL - 1
SP - 1
EP - 17
PY - 2022
DA - 2022/11
SN - 56
DO - http://doi.org/10.4208/jms.v56n1.23.01
UR - https://global-sci.org/intro/article_detail/jms/21217.html
KW - ordered tree, bush, planted tree, protected branch, unprotected branch, Catalan number, generating function.
AB - <p style="text-align: justify;">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&nbsp; branches among all branches in all trees of size $n$ approaches $ 1/3$.</p>