On bucket increasing trees, clustered increasing trees and increasing diamonds

Abstract

In this work we analyse 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 (1995, Theoret. Comput. Sci.144 221–249.) for bucket recursive trees. On the combinatorial side, we define multilabelled generalisations of the tree families d-ary increasing trees and generalised 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. (2016, Lect. Notes Comput. Sci.9644 207–219.). Concerning structural properties of bucket increasing trees, we analyse 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 analyse the parameters descendants of label j and degree of the bucket containing label j, providing distributional decompositions, complementing and extending earlier results (Kuba and Panholzer (2010), Theoret. Comput. Sci.411(34–36) 3255–3273.).

Tree evolution processes for bucket increasing trees

“…Moreover, we provide five additional equivalent properties of such evolving bucket increasing trees. This generalizes results of [23,24,27,29,31,32]. For the reader's convenience and the sake of completeness, we also collect and unify arguments of [24,27].…”

“…Tree evolution processes and combinatorial models. We collect the three growth processes generating bucket increasing trees [27,29] and the corresponding combinatorial descriptions from [24,27]. Note that here and throughout this work the capacities c(v) = c n (v) and the out-degree deg(v) = deg n (v) of a node v in a tree T are always dependent on the size |T | = n. We also mention that, from this point on, T ∈ T denotes a bucket increasing tree and not, as previously used, its unlabelled bucket ordered counterpart.…”

“…From the definitions above and the differential equation ( 3) one directly obtains the enumerative results of [24,27], which we restate below. These results are particularly of interest, as non-linear differential equations of order greater or equal three occurring in enumeration problems are extremely seldom to be solved in closed form; see for example variations of the differential equations in connection with the Blasius-type tree family [25], as well as [5].…”

“…Proposition 2 ( [24,27]). The exponential generating functions T (z) and the total weights T n of tree families generated according to Definition 1,2 and 3, respectively, are given as follows.…”

“…For the reader's convenience and the sake of completeness, we also collect and unify arguments of [24,27]. Furthermore, we generalize several results of [22,24,27] concerning the limit laws of the number of descendants of label j, providing second order results, in other words, a limit law for the number of descendants centered by its almost-sure beta limit law.…”