Space-Filling Trees

Abstract

This paper introduces the notion of space-filling trees, which are analogous to space-filling curves, but have a branching, tree-like structure and are rooted. A space-filling tree is defined by an incremental process that results in a tree for which every point in the space has a finite-length path that converges to it. In contrast to space-filling curves, individual paths in the tree are short, allowing any part of the space to be quickly reached from the root. These structures have interesting parallels in nature, including fluid distribution systems, vascular networks, and fractal plant growth, and may have many uses in engineering and computer science. We provide basic examples, general definitions and constructions, characterizations of some tree properties, and leave many open mathematical questions.