and small area.

A necessary and sufficient condition for the existence of a planar morph between two straight-line planar drawings $\Gamma_0$ and $\Gamma_1$ of a tree $T$ is that they are “topologically-equivalent”, i.e., the counter-clockwise order of the edges incident to each node $u\in V(T)$ is the same in $\Gamma_0$ and $\Gamma_1$. In order to better exploit standard terminology about tree drawings, we ensure that $\Gamma_0$ and $\Gamma_1$ are topologically-equivalent by assuming that $T$ is ordered and that $\Gamma_0$ and $\Gamma_1$ are order-preserving drawings; hence, dealing with ordered trees and with order-preserving drawings is not a loss of generality.

The width and the height of the morphs we construct are expressed not only in terms of the number of nodes of the input tree $T$, but also in terms of the width and the height of the input drawings $\Gamma_0$ and $\Gamma_1$ of $T$; this is necessary, given that $\max\{w(\Gamma_0),w(\Gamma_1)\}$ and $\max\{h(\Gamma_0),h(\Gamma_1)\}$ are obvious lower bounds for the width and the height of any morph between $\Gamma_0$ and $\Gamma_1$, respectively.

The morphs $\langle \Delta_0,\Delta_1,\dots,\Delta_k \rangle$ we construct in this paper are such that $\Delta_0,\Delta_1,\dots,\Delta_k$ are *grid* drawings, even when not explicitly specified.

In the following we introduce two tools we 