Suppose we are given a group \documentclass[12pt]{minimal}
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$\mit\Gamma $\end{document} and a tree X on which \documentclass[12pt]{minimal}
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$\mit\Gamma $\end{document} acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers \documentclass[12pt]{minimal}
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$a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \}$\end{document} if \documentclass[12pt]{minimal}
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$d\rightarrow \infty $\end{document}, where v, vo are some fixed vertices in X.¶ In this paper we consider the case where \documentclass[12pt]{minimal}
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$\mit\Gamma $\end{document} is a finitely generated group acting freely on a tree X. The growth function \documentclass[12pt]{minimal}
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$\textstyle\sum\limits a_d x^d$\end{document} is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where \documentclass[12pt]{minimal}
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$\mit\Gamma $\end{document} is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.
