Constructing Self-similar Trees with Exponential Growth

In a rooted infinite full binary tree, each vertex is the parent of exactly two children. Since there are exactly 2−1 vertices at level less than or equal to n, the infinite binary tree is said to have exponential growth with growth rate 2. Can we readily determine the growth rates of other self-similar infinite trees? We will see that the answer is yes for a class of trees that can be constructed using a method that involves repeatedly attaching copies of a finite tree to certain designated vertices. Furthermore, if an infinite tree is constructed by this method, then it has exponential growth with growth rate equal to the spectral radius of an associated nonnegative matrix. This argument exhibits a bootstrap type argument, where the general result is built up and relies on the case when the associated matrix is primitive.