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

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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.

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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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  1. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai, 200241, People’s Republic of China

    Rosena R. X. Du, Jia He & Xueli Yun

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  1. Rosena R. X. Du
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  2. Jia He
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  3. Xueli Yun
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Correspondence to Rosena R. X. Du.

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Du, R.R.X., He, J. & Yun, X. Counting Vertices with Given Outdegree in Plane Trees and k-ary Trees. Graphs and Combinatorics 35, 221–229 (2019). https://doi.org/10.1007/s00373-018-1975-8

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  • DOI: https://doi.org/10.1007/s00373-018-1975-8

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