Figure 6.

Graphical illustrations of model constructions. The top panel shows the new model constructed in Proposition 4 by joining two trees ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$. The second panel portrays the situation described in Lemma 3 and Proposition 5 and 6, in which some of the categories of $\mathcal{A}$ are replaced by ${\mathcal{B}}_k^r$. In the third panel, a new tree $\mathcal{C}$ is created by replacing every category of tree $\mathcal{A}$ by ${\mathcal{B}}^j$, j = 1, 2, ⋯ , $J_{\mathcal{A}}$, as in Proposition 7. The bottom panel demonstrates how tree ${\mathcal{B}}_{d-1}$ in Proposition 8 is created by adding the same tree structure to every category of itself recursively.