the paper is organized as follows. In \[se:preliminaries\] we present some definitions and preliminaries. In \[se:upward\] we present our results on small-area upward planar morphs between strictly-upward straight-line planar grid drawings of rooted trees. In \[se:general\] we present our main result on small-area planar morphs between straight-line planar grid drawings of trees. Finally, in \[se:conclusions\] we conclude, present some open problems, and argue about the generality of the model adopted in this paper. Namely, we prove that the problem of constructing a planar morph with polynomial resolution between two planar straight-line drawings of the same graph can be reduced to the problem of constructing a planar morph with polynomial area between two planar straight-line grid drawings of the same graph.

Preliminaries {#se:preliminaries}
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In this section we introduce some definitions and preliminaries; see also [@DETT].

#### Trees.

The node and edge sets of a tree $T$ are denoted by $V(T)$ and $E(T)$, respectively. The *degree* $\deg(v)$ of a node $v$ of $T$ is the number of its neighbors. In an *ordered* tree, a counter-clockwise order of the edges incident to each node is specified.

A *rooted tree* $T$ is a tree with one distinguished node, which is called *root* and 