Tree evolution processes for bucket increasing trees

Abstract

Bucket increasing trees are multilabelled generalizations of increasing trees, where each non-leaf node carries b labels, with b≥1 a fixed integer. We provide a fundamental result, giving a complete characterization of all families of bucket increasing trees that can be generated by a tree evolution process. We also provide several equivalent properties, complementing and extending earlier results for ordinary increasing trees to bucket trees. Additionally, we state second order results for the number of descendants of label j, again extending earlier results in the literature.