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

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.