Let $X = \gbra{a, b, c}$ $w = aaa^{-1}a^{-1}a^{-1}abb^{-1}ab^{-1}bcaa^{-1}cc^{-1}$ The reduced prefix set of $w$ is $$\redu (\prefi(w)) = \gbra{\varepsilon, a, aa, a^{-1}, b, ab^{-1}, ac, aca, acc}.$$ The Munn tree $\mt(w)$ is the following. $$ \xymatrix{ & b & {ab^{-1}} \ar[d]_{b} & \\ {a^{-1}} \ar[r]^{a} & \varepsilon \ar[u]^{b} \ar[r]^{a} & a \ar[d]^{c} \ar[r]^{a} & {aa}\\ & & {ac} \ar[r]^{c} \ar[d]^{a}& {acc}\\ & & {aca} & } $$

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