---
abstract: 'In this paper we study planar morphs between straight-line planar grid drawings of trees. A morph consists of a sequence of morphing steps, where in a morphing step vertices move along straight-line trajectories at constant speed. We show how to construct planar morphs that simultaneously achieve a reduced number of morphing steps and a polynomially-bounded resolution. We assume that both the initial and final drawings lie on the grid and we ensure that each morphing step produces a grid drawing; further, we consider both upward drawings of rooted trees and drawings of arbitrary trees.'
author:
- 'Fidel [Barrera-Cruz]{}$^1$, Manuel Borrazzo$^2$, Giordano [Da Lozzo]{}$^2$, Giuseppe $^2$, Fabrizio Frati$^2$, Maurizio Patrignani$^2$, and [Vincenzo Roselli]{}$^2$ [^1]'
bibliography:
- 'bibliography.bib'
title:  How to Morph a Tree on a Small Grid
---

Introduction {#se:introduction}
============

The problem of morphing combinatorial structures is a consolidated research topic with important applications in several areas of Computer Science such as Computational Geometry, Computer Graphics, Modeling, and Animation. The structures of interest typically are drawings of graphs; a *morph* between two drawings $\Gamma_0$ and $\Gamma_1$ of the same graph $G$ is defined as a continuously changing family of drawings $\{\Gamma_t\}$ of $G$ indexed by time $t \in 