$C^{\ast }$-algebra of the $\text{mathds}{Z}^n$-tree

Let $\mathrm{\Lambda }={\mathbb{Z}}^n$ with lexicographic ordering. $\mathrm{\Lambda }$ is a totally ordered group. Let $X={\mathrm{\Lambda }}^{+}\ast {\mathrm{\Lambda }}^{+}$. Then $X$ is a $\mathrm{\Lambda }$-tree. Analogous to the construction of graph $C^{\ast }$-algebras, we form a groupoid whose unit space is the space of ends of the tree. The $C^{\ast }$-algebra of the $\mathrm{\Lambda }$-tree is defined as the $C^{\ast }$-algebra of this groupoid. We prove some properties of this $C^{\ast }$-algebra.