Enumeration Of Highly Balanced Trees

Bereg and Wang defined a new class of highly balanced d-ary trees which they call k-trees; these trees have the interesting property that the internal path length and thus the Wiener index can be calculated quite easily. A k-tree is characterized by the property that all levels, except for the last k levels, are completely filled. Bereg and Wang claim that the number of k-trees is exponentially increasing, but do not give an asymptotic formula for it. In this paper, we study the number of d-ary k-trees and the number of mutually non-isomorphic d-ary k-trees, making use of a technique due to Flajolet and Odlyzko.