Enumeration of Protected Nodes in Motzkin Trees
Received:December 16, 2019  Revised:January 03, 2021
Key Word: Motzkin trees   protected nodes   Motzkin number   Bivariate generating function   Lagrange inversion  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11861045), Gansu Province Science Foundation for Youths (Grant No.20JR10RA187) and the Hongliu Foundation of First-Class Disciplines of Lanzhou University of Technology, China.
Author NameAffiliation
Lin YANG School of Science, Lanzhou University of Technology, Gansu 730050, P. R. China 
Shengliang YANG School of Science, Lanzhou University of Technology, Gansu 730050, P. R. China 
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Abstract:
      In this paper, we enumerate the set of Motzkin trees with $n$ edges according to the number of leaves, the number of vertices adjacent to a leaf, the number of protected nodes, the number of (protected) branch nodes, and the number of (protected) lonely nodes. Explicit formulae as well as generating functions are obtained. We also find that, as $n$ goes to infinity, the proportion of protected branch nodes and protected lonely nodes among all vertices of Motzkin trees with $n$ edges approaches $4/27$ and $2/9$.
Citation:
DOI:10.3770/j.issn:2095-2651.2021.02.002
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