is *planar* if all its intermediate drawings are planar. A morph between two strictly-upward drawings of a rooted tree is *upward* if all its intermediate drawings are strictly-upward. A morph is *linear* if each node moves along a straight-line trajectory at constant speed. Whenever the linear morph between two straight-line planar drawings $\Gamma_0$ and $\Gamma_1$ of a graph $G$ is not planar, one is usually interested in the construction of a piecewise-linear morph with small complexity between $\Gamma_0$ and $\Gamma_1$. This is formalized by defining a *morph* between $\Gamma_0$ and $\Gamma_1$ as a sequence $\langle \Gamma_0=\Delta_0,\Delta_1,\dots,\Delta_k=\Gamma_1 \rangle$ of drawings of $G$ such that the linear morph $\langle \Delta_{i-1},\Delta_{i} \rangle$ is planar, for $i=1,\dots,k$; each linear morph $\langle \Delta_{i-1},\Delta_{i} \rangle$ is called a *morphing step* or simply a *step*.

The *width* $w(\mathcal M)$ of a morph $\mathcal M=\langle \Delta_0,\Delta_1,\dots,\Delta_k \rangle$, where $\Delta_i$ is a grid drawing, for $i=0,1,\dots,k$, is equal to $\max \{w(\Delta_0),w(\Delta_1),\dots,w(\Delta_k)\}$. The *height* $h(\mathcal M)$ of $\mathcal M$ and the area of $\mathcal M$ are defined analogously.

The algorithms we design in this paper receive in input two order-preserving straight-line planar grid drawings $\Gamma_0$ and $\Gamma_1$ of an ordered tree and construct morphs $\langle \Gamma_0=\Delta_0,\Delta_1,\dots,\Delta_k=\Gamma_1 \rangle$ with few steps 