A sequential construction of a random spanning tree for the Cayley graph of a finitely generated, countably infinite subsemigroup $V$ of a group $G$ is considered. At stage $n$, the spanning tree $T$ is approximated by a finite tree $T_n$ rooted at the identity. The approximation $T_{n+1}$ is obtained by connecting edges to the points of $V$ that are not already vertices of $T_n$ but can be obtained from vertices of $T_n$ via multiplication by a random walk step taking values in the generating set of $V$. This construction leads to a compactification of the semigroup $V$ in which a sequence of elements of $V$ that is not eventually constant is convergent if the random geodesic through the spanning tree $T$ that joins the identity to the $n^{\mathrm th}$ element of the sequence converges in distribution as $n \rightarrow \infty$. The compactification is identified in a number of examples. Also, it is shown that if $h(T_n)$ and $\#(T_n)$ denote, respectively, the height and size of the approximating tree $T_n$, then there are constants $0 < c_h \le 1$ and $0 \le c_\# \le \log 2$ such that $\lim_{n \rightarrow \infty} n^{-1} h(T_n) = c_h$ and $\lim_{n \rightarrow \infty} n^{-1} \log \#(T_n) = c_\#$ almost surely.