placed in $\Delta_1$ (in $\Delta_k$) is one unit to the right of $\ell_{\Delta'_1}$ (resp. $\ell_{\Delta'_k}$), similarly as in $\Delta_2, \Delta_3, \dots, \Delta_{k-1}$, except if $l$ is to the right of every other node of $\Gamma_0$ (of $\Gamma_1$); in that case $l$ might be several units to the right of $\ell_{\Delta'_1}$ (resp. $\ell_{\Delta'_k}$). This completes the construction of $\mathcal M$. We get the following.

\[th:nary-trees-linearsteps-polyarea\] Let $T$ be an $n$-node rooted ordered tree, and let $\Gamma_0$ and $\Gamma_1$ be two [order-preserving strictly-upward straight-line planar grid]{}drawings of $T$. There exists a $(2n-1)$-step upward planar morph $\mathcal M$ from $\Gamma_0$ to $\Gamma_1$ with $h(\mathcal M) = \max\{h(\Gamma_0),h(\Gamma_1)\}$ and $w(\mathcal M) = \max\{w(\Gamma_0),w(\Gamma_1)\} +n-1$.

For a drawing $\Gamma$ of $T$ and for any two nodes $s$ and $t$ of $T$, denote by $d^v_\Gamma(s,t)$ the vertical distance between $s$ and $t$ in $\Gamma$ (that is, the absolute value of the difference between their $y$-coordinates).

We claim that the morph $\mathcal M=\langle \Gamma_0,\Delta_1,\Delta_2,\dots,\Delta_k,\Gamma_1\rangle$ defined before the statement of the theorem satisfies the requirements of the theorem and also satisfies the property that $d^v_{\Delta_i}(u,r(T))\leq \max \{d^v_{\Gamma_0}(u,r(T)),d^v_{\Gamma_1}(u,r(T))\}$, for any $i=1,2,\dots,k$ and for each node $u$ of $T$.

First, note that $\mathcal M$ has $k+1=2n-1$ steps. This is trivially 