| Enumeration of Protected Nodes in Motzkin Trees |
| Received:December 16, 2019 Revised:January 03, 2021 |
| Key Words:
Motzkin trees protected nodes Motzkin number Bivariate generating function Lagrange inversion
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| Fund Project: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. |
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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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