Transition Trees and Their Growth Rates

In this paper we define a class of infinite rooted trees called transition trees for which the growth function can be calculated using powers of a nonnegative matrix. We show that if this matrix is primitive, has spectral radius greater than 1, and does not have 1 as an eigenvalue, then the tree has exponential growth and the growth rate is the spectral radius of the matrix. We also describe a copy-and-paste method that can be used to construct any self similar infinite tree G from a finite generator tree H . We show that each generator tree H has an associated matrix whose spectral radius is the exponential growth rate of the infinite tree generated by H .