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We count the number of vertices in plane trees and $k$-ary trees with givenoutdegree, and prove that the total number of vertices of outdegree $i$ overall plane trees with $n$ edges is ${2n-i-1 \choose n-1}$, and the total numberof 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 aswell as generating function proofs.

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