Fig. 5.

Illustration of extended faces. Here $S=\{s_1,s_2\}$ corresponds to the lower- and left-face of a 2-simplex. The extended faces $\mathcal{E}(s_1)$ and $\mathcal{E}(s_2)$ are sets of vertices of ${\mathrm{Esd}}_k({\Delta }^2)$ that are on the lower- and left-boundary. The corresponding extended tree $\mathcal{E}(T)$ is the union of all these vertices. The integers indicate edge-distances ${\mathrm{dist}}_{\mathit{ET}}$ of vertices in ${\mathrm{Esd}}_k({\Delta }^2)$ from $\mathcal{E}(T)$