measured by comparing the area of $\Gamma_0$ and $\Gamma_1$ with the area of the $\Delta_i$’s. We remark that morphs between planar orthogonal drawings of maximum-degree-$4$ planar graphs, like those in [@DBLP:journals/talg/BiedlLPS13; @DBLP:conf/compgeom/GoethemV18], inherently have polynomial resolution.

In this paper we show how to construct morphs of tree drawings that simultaneously achieve a reduced number of morphing steps and a polynomially-bounded resolution. Adopting the setting of [@DBLP:conf/gd/Barrera-CruzHL14], we assume that $\Gamma_0$ and $\Gamma_1$ are grid drawings and we ensure that each morphing step produces a grid drawing.

We present three algorithms. The first two algorithms construct morphs between any two strictly-upward straight-line planar grid drawings $\Gamma_0$ and $\Gamma_1$ of $n$-node rooted trees; *strictly-upward* drawings are such that each node lies above its children. Both algorithms construct morphs in which each intermediate grid drawing has linear width and height, where the input size is measured by $n$ and by the width and the height of $\Gamma_0$ and $\Gamma_1$. The first algorithm employs $\Theta(n)$ morphing steps. The second algorithm employs $\Theta(1)$ morphing steps, however it only applies to binary trees. The third algorithm allows us to achieve our main result, namely that for any two straight-line planar grid drawings $\Gamma_0$ and $\Gamma_1$ of 