Copy-paste trees and their growth rates

Abstract

In this paper, we describe a copy-and-paste method for constructing a class of infinite self-similar trees. A copy-paste tree is constructed by repeatedly attaching copies of a finite tree (called a \textit {generator}) to certain designated attachment vertices. We show that each generator has an associated nonnegative matrix which can be used to determine a formula for the growth function of the copy-paste tree. In our main theorem, we use results from Perron-Frobenius theory to show that every copy-paste tree has exponential growth, with growth rate equal to the spectral radius of its associated matrix.