In this work we study edge weights for two specific families of increasing trees, which include binary increasing trees and plane oriented recursive trees as special instances, where plane-oriented recursive trees serve as a combinatorial model of scale-free random trees given by the m = 1 case of the BarabásiAlbert model. An edge e = (k, l), connecting the nodes labeled k and l, respectively, ...

Bona [6] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [14]. Recently, Janson [18] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier resu...

Simply generated families of trees are described by the equation T (z) = φ(T (z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label ∈ {1, . . . , n}, no label occurs twice, and whenever we proceed from the root to a leaf, the labels are increasing. This leads to the concept of simple families of increasing trees. Three such families are ...

A tree is called k-decomposable if it has a spanning forest whose components are all of size k. In this paper, we study the number of k-decomposable trees in families of increasing trees, i.e. labeled trees in which the unique path from the root to an arbitrary vertex forms an increasing sequence. Functional equations for the corresponding counting series are provided, yielding asymptotic or ev...

Abstract. Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i. e. labellings of the nodes by distinct integers of the set {1, . . . , n} in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular intere...

By a theorem of Dobrow and Smythe, the depth of the kth node in very simple families of increasing trees (which includes, among others, binary increasing trees, recursive trees and plane ordered recursive trees) follows the same distribution as the number of edges of the form j−(j+1) with j < k. In this short note, we present a simple bijective proof of this fact, which also shows that the resu...

Learning from unbalanced datasets presents a convoluted problem in which traditional learning algorithms typically perform poorly. The heuristics used in learning tend to favor the larger, less important classes in such problems. While other methods, like sampling, have been introduced to combat imbalance, these tend to be computationally expensive. This paper proposes Hellinger distance as a m...

In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in the tree can get multiple labels. For all tree classes we show characterizations of suitable generating functions for the tree enumeration sequence via diff...