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

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 $\left(\genfrac{}{}{0px}{}{2n-i-1}{n-1}\right)$, and the total number of vertices of outdegree $i$ over all $k$-ary trees with $n$ edges is $\left(\genfrac{}{}{0px}{}{k}{i}\right)\left(\genfrac{}{}{0px}{}{kn}{n-i}\right)$. For both results we give bijective proofs as well as generating function proofs.