The vertices of the Cayley graph of a finitely generated semigroup form a set
    of sites which can be labeled by elements of a finite alphabet in a manner
    governed by a nonnegative real interaction matrix, respecting nearest neighbor
    adjacency restrictions. To the set of these configurations one can associate a
    pressure, which is defined as the limit, when it exists, of averages of the
    logarithm of the partition function over certain finite subgraphs. We prove
    that for shifts of finite type on generalized Fibonacci trees, under an added
    condition, the limit exists and is given by an infinite series. We also show
    that the limit of any cluster points of the pressure on finite subtrees as the
    number of generators grows without bound, which we call the asymptotic
    pressure, equals the logarithm of the maximum row sum of the interaction

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