[Paper Review] Counting Rooted Trees: The Universal Law t(n) ~ C ρ^{-n} n^{-3/2}

Jason P. Bell, Stanley Burris+1 more|arXiv (Cornell University)|Dec 19, 2005

Advanced Combinatorial Mathematics15 references4 citations

TL;DR

This paper establishes a universal asymptotic law for the number of rooted trees in combinatorial classes recursively defined by standard constructions such as sequence, multiset, and cycle. Using complex analysis and singularity theory, it proves that the number of such trees, t(n), asymptotically follows t(n) ~ Cρ^(-n)n^(-3/2), where ρ is the radius of convergence of the generating function and C is a constant, extending Pólya’s classical result to a broad class of recursively defined tree families.

ABSTRACT

Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence; for most of these a simple test can be used to quickly show that the form of the asymptotics is the same as that for the class of rooted trees: C ρ^{-n} n^{-3/2} where ρis the radius of convergence of T.

Motivation & Objective

• To establish a universal asymptotic form t(n) ~ Cρ^(-n)n^(-3/2) for a broad class of recursively defined rooted tree families.
• To generalize Pólya’s and Odlyzko’s results on tree generating functions to include combinations of standard combinatorial constructions.
• To provide a unified framework for asymptotic enumeration of rooted trees using analytic combinatorics and singularity analysis.
• To investigate whether the universal law holds when the set operator (Set) is included in the recursive constructions.
• To explore the extension of the asymptotic law to systems of equations and monadic second-order definable classes of trees.

Proposed method

• Apply the implicit function theorem and Weierstrass preparation theorem to analyze the dominant singularity of generating functions defined by recursive equations.
• Use the Cauchy integral formula to derive asymptotics from the location and type of the dominant singularity.
• Characterize the radius of convergence ρ and the constant C in terms of the solution to the functional equation T(z) = E(z, T(z)).
• Employ singularity analysis techniques to show that the singularity at ρ is of square-root type, leading to the n^(-3/2) decay factor.
• Introduce the class DOM[z,w] of functions analytic in a polydisk and use spectral calculus to analyze periodic and multi-singularity cases.
• Apply the method of dominant singularity analysis to systems of equations defining constructible classes of trees.

Experimental results

Research questions

• RQ1 Does the universal asymptotic law t(n) ~ Cρ^(-n)n^(-3/2) hold for recursively defined tree classes constructed using the set operator (Set) in addition to standard constructions?
• RQ2 If a generating function T(z) has a shift-periodic form z^d V(z^q), does the reduced system V(z) = H(z, V(z)) inherit the same singularity properties and asymptotic law?
• RQ3 Can the asymptotic law be extended to systems of equations T_i(z) = Θ_i(T_1, ..., T_k) defining constructible classes of trees?
• RQ4 For monadic second-order definable classes of trees, does the universal law hold on each component of the finite decomposition of such classes?
• RQ5 What is the asymptotic behavior of free tree classes obtained by removing the root from recursively defined rooted tree classes?

Key findings

• The asymptotic form t(n) ~ Cρ^(-n)n^(-3/2) universally applies to all combinatorial classes of rooted trees defined by recursive combinations of sequence, multiset, cycle, and directed cycle constructions.
• The radius of convergence ρ is the unique positive solution to the equation E_w(ρ, T(ρ)) = 1, where E(z,w) is the generating function operator.
• The singularity at ρ is of square-root type, confirmed by non-vanishing second derivatives E_z and E_ww, leading to the n^(-3/2) factor.
• The constant C is explicitly determined by the first and second derivatives of E at the dominant singularity (ρ, T(ρ)).
• The result extends to planar trees and binary trees, with specific examples confirming the law holds for both.
• The paper shows that the universal law persists even when the generating function has a shift-periodic structure, provided the reduced system satisfies the required analytic conditions.