Counting vertices in plane and $k$-ary trees with given outdegree

Rosena R. X. Du; Jia He; Xueli Yun

Counting vertices in plane and $k$-ary trees with given outdegree

Combinatorics 2019-03-19 v3

Authors: Rosena R. X. Du , Jia He , Xueli Yun

Abstract

We count the number of vertices in plane trees and $k$ -ary trees with given outdegree, and prove that the total number of vertices of outdegree $i$ over all plane trees with $n$ edges is ${2n-i-1 \choose n-1}$ , and the total number of vertices of outdegree $i$ over all $k$ -ary trees with $n$ edges is ${k\choose i}{kn\choose n-i}$ . For both results we give bijective proofs as well as generating function proofs.

Keywords

plane trees k-ary trees outdegree vertex counting bijective proofs generating functions combinatorics

Cite

@article{arxiv.1501.07468,
  title  = {Counting vertices in plane and $k$-ary trees with given outdegree},
  author = {Rosena R. X. Du and Jia He and Xueli Yun},
  journal= {arXiv preprint arXiv:1501.07468},
  year   = {2019}
}

Comments

8 pages, 3 figures, deleted generating function proofs of the two main theorems, and added Theorem 3.9

Counting vertices in plane and $k$-ary trees with given outdegree

Abstract

Keywords

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