Using linear forms to determine the set of integers realizable by \((g_ 0,g_ 1,\dots ,g_ n)\)-trees
The paper deals with the problem, how many leaves a tree can have whose internal nodes all have outvalences (number of immediate successors) out of \(g_ 0,g_ 1,...,g_ n\) and all of whose leaves are on the same level. Of particular interest is the conductor which is the least number, such that all \(N\geq k\) can occur as numbers of leaves for such trees. Since the number of leaves can be counted by \(1+(g_ 0-1)x_ 0+...+(g_ n-1)x_ n,\) where \(x_ i\) is the number of nodes of outvalence \(g_ i\), the integers in question are in intimate connection to the values the linear form \(\sum (g_ i-1)x_ i\) takes. Large intervals in the set of solutions are determined and bounds for the conductor are given.
scientific article; zbMATH DE number 3873353