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k-Protected Vertices in Unlabeled Rooted Plane Trees

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Abstract

We find a simple, closed formula for the proportion of vertices which are k-protected in all unlabeled rooted plane trees on n vertices. We also find that, as n goes to infinity, the average rank of a random vertex in a tree of size n approaches 0.727649, and the average rank of the root of a tree of size n approaches 1.62297.

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Acknowledgements

I am grateful to Miklos Bóna for guidance on the ins and outs of the submission process, as well as careful reading of the manuscript; my wife, Jaclyn van Wingerden for her many readings and constant support; Arnold Knopfmacher and Aubrey Blecher for their very helpful comments; and the referees for helpful suggestions.

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  1. Department of Mathematics, University of Florida, Gainseville, FL, 32612, USA

    Keith Copenhaver

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  1. Keith Copenhaver
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Correspondence to Keith Copenhaver.

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Copenhaver, K. k-Protected Vertices in Unlabeled Rooted Plane Trees. Graphs and Combinatorics 33, 347–355 (2017). https://doi.org/10.1007/s00373-017-1772-9

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  • DOI: https://doi.org/10.1007/s00373-017-1772-9

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