On the number of increasing trees with label repetitions

doi:10.1016/j.disc.2019.111722

Discrete Mathematics

Volume 343, Issue 8, August 2020, 111722

https://doi.org/10.1016/j.disc.2019.111722 Get rights and content

Abstract

We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the same label can appear in distinct branches of the tree.

In the course of the analysis we develop a method to extract asymptotic information on the coefficients of purely formal power series. The method is based on an approximate Borel transform (or, more generally, Mittag-Leffler transform) which enables us to quickly guess the exponential growth rate. With this guess the sequence is then rescaled and a singularity analysis of the generating function of the scaled counting sequence yields accurate asymptotics. The actual analysis is based on differential equations and a Tauberian argument.

The counting problem for trees of size $n$ exhibits interesting asymptotics involving powers of $n$ with irrational exponents.

Introduction

A rooted binary plane tree of size $n$ (meaning that it has $n$ vertices) is called monotonically increasingly labeled with the integers in ${1, 2, \dots, k}$ if the vertices of the tree are labeled with those integers and each sequence of labels along a path from the root to any leaf is weakly increasing. The concept of a monotonically labeled tree (of fixed arity $t \geq 2$ ) has been introduced in the 1980s by Prodinger and Urbanek [11] and has then been revisited by Blieberger [1] in the context of Motzkin trees. In the latter paper, monotonically labeled Motzkin trees are directly related to the enumeration of expression trees that are built during compilation or in symbolic manipulation systems.

A rooted binary plane tree of size $n$ is called increasingly labeled, if it is monotonically increasingly labeled with the integers in ${1, 2, \dots, n}$ and each integer from $1$ to $n$ appears exactly once. In particular, this implies that the sequences of labels along the branches are strictly increasing. This model corresponds to the heap data structure in computer science and is also related to the classical binary search tree model.

In this paper we are interested in a model lying between the two previous ones. It is in fact a special subclass of monotonically labeled increasing trees: each sequence of labels from the root to any leaf is strictly increasing and each integer between $1$ and $k$ must appear in the tree, where $k$ is the largest label. The main difference from the classical model of binary increasing trees (cf. e.g. the book of Drmota [5]) is that the same label can appear in distinct branches of the tree. Our interest in such a model arises from the following fact: There is a classical evolution process, presented for example in [5], to grow a binary increasing tree by replacing at each step an unlabeled leaf by an internal node labeled by the step number and attached to two new leaves. Here, we extend the process by selecting at each step a subset of leaves and replacing each of them by the same structure (the labeled internal nodes, all with the same integer label, and their two children), thus an increasing binary tree with repetitions is under construction.

Such a model may serve to describe population evolution processes where each individual can give birth to two descendants independently of the other individuals. In another paper [4] we have presented an increasing model with repetitions of Schröder trees that encodes the chronology in phylogenetic trees.

Finally, by merging the nodes with the same label we obtain directed acyclic graphs whose nodes are increasingly labeled (without repetitions). Such an approach introduces a new model of concurrent processes with synchronization that induces processes whose description is more expressive than the classical series–parallel model that we have studied in [2], [3].

Another main feature of this paper is the methodological aspect. While we are “only” presenting a first order asymptotic analysis of the counting sequence, we introduce an approach to deal with generating functions which are on the one hand given by some nontrivial functional equation, on the other hand purely formal power series.

Our approach falls into the guess-and-prove paradigm, which is frequently used in algebraic combinatorics and partially automatized there. We will apply an approximate Borel transform to obtain heuristically an equation for the transformed generating functions. Then we use a scaling indicated by the heuristics in order to deal with a moderately growing sequence. The generating function of the scaled sequence then admits an asymptotic analysis which leads to the asymptotic evaluation of the sequence, including a proof of the guessed result after all.

In Section 2 we introduce the concept of increasing binary trees with repetitions. In particular, we develop the evolution process naturally defining such trees and present the asymptotic behavior of its enumeration sequence.

Section 3 is devoted to the asymptotic study of the number of increasing binary trees with repetitions of size $n$ . In the first instance, we partition the problem and get some recurrence relation. Then we present the methodology based on the approximate Borel transform. Afterwards, we derive the functional equation from the evolution process and then analyze the counting sequence, first heuristically and, after having gained the insight from the heuristics, then exactly.

Then, in Section 4 we present a brief discussion of the generalization to $k$ -ary trees.

Section snippets

Basic concepts and statement of the main result

The concept of an increasing tree is well studied in the literature (cf. for example [5]). An increasing tree is defined as a rooted labeled tree where on each path from the root to a leaf the sequence of labels is increasing. In fact they are strictly increasing, since the nodes of a labeled tree with $n$ nodes carry exactly the labels $1, 2, \dots, n$ . The aim of the paper is to introduce a weaker model of increasing trees where repetitions of the labels can appear.

Definition 1

A weakly increasing binary tree is

•
a

Enumeration of weakly increasing binary trees

Using the combinatorial evolution process to build trees (described in the previous section) and its associated Lemma 2, we get directly a recurrence for the partition of the set of all weakly increasing trees of size $n$ according to their maximal label $m$ : $B_{1, 2} = 1$ $B_{1, n} = 0 if n \neq 2$ $B_{m, n} = \sum_{ℓ = 1}^{n - m + 2} (\binom{n - ℓ}{ℓ}) B_{m - 1, n - ℓ},$ where $B_{m, n}$ is the number of weakly increasing trees with $n$ nodes in which exactly $m$ distinct labels occur. We remark that $B_{n} = \sum_{m \geq 1} B_{m, n}$ , and thus the first terms of ${(B_{n})}_{n \geq 0}$ are $0, 0, 1, 2, 7, 34, 214, 1652,$

Higher arity weakly increasing trees

In this section we briefly discuss how our results generalize to $k$ -ary weakly increasing trees with repetitions. The definitions about the binary case can be adapted in an obvious way. The size of the structures corresponds to the number of leaves of the completed $k$ -ary tree and the generating function $G (z) = \sum_{n \geq k} G_{n} z^{n}$ where $G_{n}$ is the number of $k$ -ary weakly increasing trees (with repetitions) of size $n$ . In a similar way as in the binary case we obtain the functional equation $G (z) = \frac{1}{2} (z^{k} + G (z + z^{k})) .$ We

Conclusion

In this article, we have shown that the asymptotic behavior of weakly increasing binary trees of size $n$ is given by $B_{n} \underset{n \to \infty}{\sim} η n^{- ln 2} {(\frac{1}{ln 2})}^{n} (n - 1)!$ where $η$ is a constant. This exhibits a certain oddity compared to the classic asymptotic behavior of trees. In particular, the presence of the polynomial factor $n^{- ln (2)}$ is quite unusual and can be compared to the classical factor $n^{- 3 ∕ 2}$ for the simple family of trees.

To keep the presentation concise, we have performed only asymptotics up to the first order.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors thank the anonymous referees for their comments and suggested improvements.

References (13)

BliebergerJ.
Monotonically labelled Motzkin trees
Discrete Appl. Math.
(1987)
ProdingerH. et al.
On monotone functions of tree structures
Discrete Appl. Math.
(1983)
O. Bodini, M. Dien, A. Genitrini, F. Peschanski, The ordered and colored products in analytic combinatorics:...
O. Bodini, M. Dien, A. Genitrini, F. Peschanski, Entropic uniform sampling of linear extensions in series-parallel...
BodiniO. et al.
Ranked Schröder trees
DrmotaM.
Random Trees
(2009)

There are more references available in the full text version of this article.

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^☆: This work was partially supported by the ANR project MetACOnc ANR-15-CE40-0014, by the Austrian Science Fund (FWF) grant SFB F50-03, the PHC Amadeus project 39454SF, and the National Research Foundation of South Africa , grant number 96236.

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On the number of increasing trees with label repetitions☆