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

摘要

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 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 ki kn n- i . For all these results we give bijective proofs.