Before revision:  To be precise, we first devise an algorithm that takes O(n) morphing steps but uses a grid of small volume. We then use this algorithm to devise a better algorithm that takes \mathcal{O}(\sqrt{n} \log n) morphing steps.
After revision:  To be precise, we first devise an algorithm that takes O(n) morphing steps but uses a grid of polynomial size. Can the number of steps in this setting be reduced substantially by using 3D? We answer this question in affirmative and present a 3D non-crossing morph between two planar grid drawings of an n-vertex tree in \mathcal{O}(\sqrt{n} \log n) morphing steps.