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Graphs and Combinatorics

, Volume 35, Issue 1, pp 221–229 | Cite as

Counting Vertices with Given Outdegree in Plane Trees and k-ary Trees

  • Rosena R. X. DuEmail author
  • Jia He
  • Xueli Yun
Original Paper
  • 41 Downloads

Abstract

We count the number of vertices with given outdegree in plane trees and k-ary trees, and get the following results: the total number of vertices of outdegree i among all plane trees with n edges is \({2n-i-1 \atopwithdelims ()n-1}\); the total number of vertices of degree i among all plane trees with n edges is twice this number; and the total number of vertices of outdegree i among all k-ary trees with n edges is \({k\atopwithdelims ()i}{kn\atopwithdelims ()n-i}\). For all these results we give bijective proofs.

Keywords

Plane trees k-ary trees Outdegree Degree Composition Bijective proof 

Mathematics Subject Classification

05A15 05C05 05C07 

Notes

Acknowledgements

This work is partially supported by National Natural Science Foundation of China (No. 11871223) and the Science and Technology Commission of Shanghai Municipality (No. 18ZR1411700 and No. 18dz2271000).

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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Shanghai Key Laboratory of PMMPEast China Normal UniversityShanghaiPeople’s Republic of China

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