Level of nodes in increasing trees revisited

Abstract

ABSTRACT: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 especially important: recursive trees, heap ordered trees, and binary increasing trees. They belong to the subclass of very simple families of increasing trees, which can be characterized in 3 different ways.This paper contains results about these families as well as about polynomial families (the function ϕ(u) is just a polynomial). The random variable of interest is the level of the node (labelled) j, in random trees of size n ≥ j. For very simple families, this is independent of n, and the limiting distribution is Gaussian. For polynomial families, we can prove this as well for j, n → ∞ such that n−j is fixed. Additional results are also given. These results follow from the study of certain trivariate generating functions and Hwang's quasi power theorem. They unify and extend earlier results by Devroye, Mahmoud, and others.

Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

“…Considering the above insertion rule, the parent u of the nth internal node is chosen with probability proportional to b − deg(u), where deg(u) is the number of internal children of node u in the tree with n − 1 nodes and each of the deg(u) + 1 possible positions for the new node are equally likely. In Panholzer and Prodinger (2007) and Kuba and Panholzer (2010) it was shown that this tree is the same as the b-ary increasing tree, which belongs to the simple families of increasing trees introduced in Bergeron et al (1992). In Drmota (2009, Section 1.3.3) this tree is also called the b-ary recursive tree.…”

The Subtree Size Profile of Plane-oriented Recursive Trees

“…It is interesting to ask which other simple classes of increasing trees admit such a construction (via a natural tree evolution process). This question was solved in Panholzer and Prodinger [24], where it was shown that up to scaling such a construction exists if and only if the class belongs to following three types.…”

The Height of Increasing Trees

“…In a recent paper Panholzer and Prodinger [12] proved that there are exactly three families where the sequence P n of probability measures on J n is induced by a (natural) tree evolution process (described below) if and only if Ψ(t) has one of the three forms:…”

“…Note that for every scaling factor c > 0 the function F (c y) is also a solution of (12). Thus, we can assume (without loss of generality) that there is F α that satisfies (12) and…”

“…Next we will define auxiliary functions y k (α, z) and Y k (α, z). For this purpose we have to solve an integral equation that is similar to (12). We omit the proof.…”