On bucket increasing trees, clustered increasing trees and increasing diamonds

Abstract

In this work we analyze bucket increasing tree families. We introduce two simple stochastic growth processes, generating random bucket increasing trees of size n, complementing the earlier result of Mahmoud and Smythe [15] for bucket recursive trees. On the combinatorial side, we define multilabelled generalizations of the tree families d-ary increasing trees and generalized plane-oriented recursive trees. Additionally, we introduce a clustering process for ordinary increasing trees and relate it to bucket increasing trees. We discuss in detail the bucket size two and present a bijection between such bucket increasing tree families and certain families of graphs called increasing diamonds, providing an explanation for phenomena observed by Bodini et al. [3].Concerning structural properties of bucket increasing trees, we analyze the tree parameter K n . It counts the initial bucket size of the node containing label n in a tree of size n and is closely related to the distribution of node types. Additionally, we analyze the parameters descendants of label j and degree of the bucket containing label j, providing distributional decompositions, complementing and extending earlier results [10].