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

TL;DR

This paper derives formulas for counting vertices with specific outdegree in plane and $k$-ary trees, providing bijective and generating function proofs for these combinatorial results.

Contribution

It introduces explicit formulas for the total number of vertices with a given outdegree in plane and $k$-ary trees, with bijective and generating function proofs.

Findings

Total vertices of outdegree i in plane trees: ${2n-i-1 race n-1}$

Total vertices of outdegree i in $k$-ary trees: ${k race i}{kn race n-i}$

Provides bijective and generating function proofs for these formulas.

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.