Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equivalence Classes Induced by Binary Tree Isomorphism -- Generating Functions

David Serena, William J Buchanan

TL;DR

This work investigates equivalence classes arising from binary-tree isomorphism and develops a unifying generating-function framework to enumerate rooted binary trees under various isomorphism constraints and colorings. It begins with the classic ordered-binary-tree Catalan enumeration, then advances to non-isomorphic rooted trees via a multivariate generating functions approach, introducing counts $K_{n,\ell}$ and a master equation $M(x,y)=1+\frac{x(1+y)}{2}[M(x,y)^2+M(x^2,y^2)]$. Building on this, it extends to two-color trees with counts $B_{n,k}$, and finally to triple-parameter counts $K_{n,\ell,c}$ with generating function $S(x,y,z)$ obeying $S(x,y,z)=1+\frac{x(1+z)}{2}\bigl[(2-y)S(x^2,y^2,z^2)+yS(x,y,z)^2\bigr]$, showing how these multivariate equations reduce to known forms under specialization. The results provide exact, multiplicatively separable generating-function formulations for enumerating isomorphism classes across node counts, colorings, and sibling-structure constraints, enabling efficient computation and deeper combinatorial insight into binary-tree isomorphism classes.

Abstract

Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree \cite{abboud2018subtree}. Binary tree isomorphism is an important problem in computer science. The enumeration of the number of non-isomorphic rooted binary trees is therefore well known. The paper reiterates the known results for ordered binary trees and presents previous results for the enumeration of non-isomorphic rooted binary trees. Then, new enumeration results are put forward for two-colour binary tree isomorphism parametrized by the number of nodes, the number of specific colours and the number of non-isomorphic sibling subtrees. Multi-variate generating function equations are presented that enumerate these tree structures. The generating functions with these parametrizations separate multiplicatively into simplified generating function equations.

Equivalence Classes Induced by Binary Tree Isomorphism -- Generating Functions

TL;DR

This work investigates equivalence classes arising from binary-tree isomorphism and develops a unifying generating-function framework to enumerate rooted binary trees under various isomorphism constraints and colorings. It begins with the classic ordered-binary-tree Catalan enumeration, then advances to non-isomorphic rooted trees via a multivariate generating functions approach, introducing counts and a master equation . Building on this, it extends to two-color trees with counts , and finally to triple-parameter counts with generating function obeying , showing how these multivariate equations reduce to known forms under specialization. The results provide exact, multiplicatively separable generating-function formulations for enumerating isomorphism classes across node counts, colorings, and sibling-structure constraints, enabling efficient computation and deeper combinatorial insight into binary-tree isomorphism classes.

Abstract

Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree \cite{abboud2018subtree}. Binary tree isomorphism is an important problem in computer science. The enumeration of the number of non-isomorphic rooted binary trees is therefore well known. The paper reiterates the known results for ordered binary trees and presents previous results for the enumeration of non-isomorphic rooted binary trees. Then, new enumeration results are put forward for two-colour binary tree isomorphism parametrized by the number of nodes, the number of specific colours and the number of non-isomorphic sibling subtrees. Multi-variate generating function equations are presented that enumerate these tree structures. The generating functions with these parametrizations separate multiplicatively into simplified generating function equations.

Paper Structure

This paper contains 11 sections, 101 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Basic Ordered Binary Trees
  3. Non-Isomorphic Binary Trees With Labeled Root Node
  4. Generating Functions
  5. Equation in G(s)
  6. Summary of Generating Function Formulas
  7. Number of Non-Isomorphic Rooted Binary Trees with Number of Non-Isomorphic Siblings Given
  8. Bivariate Generating Function for $K_{n,\ell}$ Recurrence
  9. Summary of System of Functional Equations
  10. 2-colored Rooted Non-isomorphic Binary Trees
  11. 2-Color Rooted Binary Tree Isomorphism, Parameritized with Number of Specific Color, Non-Isomorphic Siblings and Nodes

Figures (5)

  • Figure 1: Rooted Non-Isomorphic Binary Trees
  • Figure 2: Rooted Binary Trees, Parameritized with Number of Nodes=$n$, and Number of Non-Isomorphic Siblings=$\ell$
  • Figure 3: 2-Color Binary Trees, Parametrized by Number of Nodes=$n$ and Number of a Specific Color=$k$
  • Figure 4: Rooted Binary Trees, Parameritized with Number of Nodes=$n$, Number of Non-Isomorphic Siblings=$\ell$ and Number of Specific Color=$c$
  • Figure 5: Continued: Rooted Binary Trees, Parameritized with Number of Nodes=$n$, Number of Non-Isomorphic Siblings=$\ell$ and Number of Specific Color=$c$