with $v$; this is indeed true for $t=0$ and $t=1$. Suppose that, for some $t\in (0,1)$, the left-to-right order in which two edges $(u_1,v)$ and $(u_2,v)$ enter $v$ in $\Gamma_t$ is different than in $\Gamma_0$. Since such edges are represented by curves monotonically increasing in the $y$-direction from $u_1$ and $u_2$ to $v$ throughout $\mathcal M$, it follows that there is a time $t^*\in (0,t)$ such that the edges $(u_1,v)$ and $(u_2,v)$ overlap in $\Gamma_{t^*}$. However, this cannot happen due to the planarity of $\mathcal M$.

Upward Planar Morphs of Rooted-Tree Drawings {#se:upward}
============================================

In this section we show how to construct small-area morphs between order-preserving strictly-upward straight-line planar grid drawings of rooted ordered trees.

Our first result shows that such morphs can always be constructed consisting of a linear number of steps. This is obtained via an inductive algorithm which is described in the following. Let $T$ be an $n$-node rooted ordered tree. The *rightmost path* of $T$ is the maximal path $(s_0,\dots,s_m)$ such that $s_0=r(T)$ and $s_{i}$ is the rightmost child of $s_{i-1}$, for $i=1,\dots,m$. Note that $s_m$ is a leaf, which is called the *rightmost leaf* $l^{\rightarrow}_T$ of $T$. For a straight-line grid drawing $\Gamma$, denote by 