Collection Algebra

Since much of rooted tree generation seems to be nowadays computer-based (especially the tree generation algorithms), this book aims to codify rooted tree generation processes algebraically, construct a formal theory for rooted tree generation, and extend the applicability of this theory to sum form equations, a new area of research.

A rooted tree generation process can be defined with the iterative use of a successor operator, or with the use of a generator operator, and many classic rooted tree generation problems can be reformulated as solving collection algebraic equations. This naturally leads to a more generalized scene where not only generation processes are investigated, but also questions such as “What is the successor of a specific rooted tree r?” and “If I take a successor of an unknown rooted tree r and connect two leaves to the root of the tree successor Sr, and the result is the root with six leaves, then what is r?”. The questions are stated in algebraic form, in other words as equations. The scope of rooted tree generation theory is thus extended to sum form equations, or collection equations.

This book is intended for experts, researchers, scholars, doctoral and graduate students in graph theory, and especially rooted tree theory and generation. The theory also has various connections to other branches of mathematics, such as formal languages, abstract algebra, combinatorics, and number theory, and as such, mathematicians working in these branches may also be interested. One possible application area is also computer science, and thus, the content of the book may also be useful and interesting to computer scientists.