Next, consider any index $i\in \{1,2,\dots,k\}$ and any node $u$ of $T$. We prove that $d^v_{\Delta_i}(u,r(T))\leq \max \{d^v_{\Gamma_0}(u,r(T)),d^v_{\Gamma_1}(u,r(T))\}$.

-   If $u\neq l$, then we have $d^v_{\Delta_i}(u,r(T))= d^v_{\Delta'_i}(u,r(T'))$. By induction, we have $d^v_{\Delta'_i}(u,r(T'))\leq \max \{d^v_{\Gamma'_0}(u,r(T')),d^v_{\Gamma'_1}(u,r(T'))\}$. Since $d^v_{\Gamma'_0}(u,r(T'))= d^v_{\Gamma_0}(u,r(T))$ and $d^v_{\Gamma'_1}(u,r(T'))= d^v_{\Gamma_1}(u,r(T))$, we have $d^v_{\Delta_i}(u,r(T))\leq \max \{d^v_{\Gamma_0}(u,r(T)),d^v_{\Gamma_1}(u,r(T))\}$ as required.

-   By construction, we have $d^v_{\Delta_i}(l,r(T))= d^v_{\Delta_i}(\pi,r(T))+1$. Further, $d^v_{\Delta_i}(\pi,r(T))=d^v_{\Delta'_i}(\pi,r(T'))$. By induction we have $d^v_{\Delta'_i}(\pi,r(T'))\leq \max \{d^v_{\Gamma'_0}(\pi,r(T')),d^v_{\Gamma'_1}(\pi,r(T'))\}$. Since $d^v_{\Gamma'_0}(\pi,r(T'))=d^v_{\Gamma_0}(\pi,r(T))$ and $d^v_{\Gamma'_1}(\pi,r(T'))=d^v_{\Gamma_1}(\pi,r(T))$, we have $d^v_{\Delta_i}(l,r(T)) \leq \max \{d^v_{\Gamma_0}(\pi,r(T)),d^v_{\Gamma_1}(\pi,r(T))\}+1$. Since $\Gamma_0$ and $\Gamma_1$ are strictly-upward, we have $d^v_{\Gamma_0}(l,r(T))\geq d^v_{\Gamma_0}(\pi,r(T))+1$ and $d^v_{\Gamma_1}(l,r(T))\geq d^v_{\Gamma_1}(\pi,r(T))+1$, hence $d^v_{\Delta_i}(l,r(T)) \leq \max \{d^v_{\Gamma_0}(l,r(T)),d^v_{\Gamma_1}(l,r(T))\}$ as required.

Since $d^v_{\Delta_i}(u,r(T))\leq \max \{d^v_{\Gamma_0}(u,r(T)),d^v_{\Gamma_1}(u,r(T))\}$, for any index $i\in \{1,2,\dots,k\}$ and any node $u$ of $T$, we directly get $h(\Delta_i)\leq \max\{h(\Gamma_0),h(\Gamma_1)\}$ and hence $h(\mathcal M) = \max\{h(\Gamma_0),h(\Gamma_1)\}$.

It remains to prove the planarity of $\mathcal M$; then Lemma \[le:upwardplanar-implies-order\] implies that the drawing of $T$ is order-preserving throughout $\mathcal M$. The following property is useful (refer to \[fig:upward-linearsteps-polyarea-property\]).

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\[pr:empty-right\] Consider any [order-preserving strictly-upward straight-line planar grid]{}drawing $\Gamma$ of an ordered rooted tree $S$. Let $h^1_{\Gamma}$ and $h^2_{\Gamma}$ be the horizontal half-lines starting at the root $r(S)$ and at the rightmost leaf $l^{\rightarrow}_S$ of $S$, respectively, and directed rightwards. Let $\mathcal R_{\Gamma}$ be the region of the plane 