Using linear forms to determine the set of integers realizable by ( g0, g1, …, gn)-trees

Abstract

A ( g 0, g 1, …, g n )-tree is a tree whose internal nodes have outvalences g 0, g 1, …, g n−1 , or g n and whose leaves are all on the same level. It has been shown that all but a finite set of integers may be realized as the number of leaves on a ( g 0, …, g n )-tree if and only if the gcd ( g 1 - g 0, … g n - g 0)=1 In this paper properties of the set of realizable integers are discussed including upper and lower bounds for the conductor κ (the conductor is the least integer having the property that for all N ⩾ κ, N is realizable). It is also noted that several problems, including finding the set of realizable integers for ( g 0, … g n )-AVL trees and (1, g 1, …, g n )-brother trees, are all equivalent to the problem of Frobenius concerning the assumed values of a linear form in nonnegative integers.