Enumeration of Highly Balanced Trees

Abstract

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.