example of Munn tree

Let $X=\{a,b,c\}$, $w=aa{a}^{-1}{a}^{-1}{a}^{-1}ab{b}^{-1}a{b}^{-1}bca{a}^{-1}c{c}^{-1}$. The reduced prefix set of $w$ is

$$\mathrm{red}(\mathrm{pref}(w))=\{\epsilon ,a,aa,a^{-1},b,a{b}^{-1},ac,aca,acc\}.$$

The Munn tree $\mathrm{MT}(w)$ is the following.

Note that we have drawn only edges of the form $(v_1,x,v_2)$ (i.e. ) with $x\in X$, leaving implicit the existence of the opposite edges $(v_2,x^{-1},v_1)$ (i.e. ), as usual in the diagram representation of inverse word graphs.