Space-filling tree

Space-filling trees are geometric constructions that are analogous to space-filling curves,^[1] 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.^[2][3] The simplest examples of space-filling trees have a regular, self-similar, fractal structure, but can be generalized to non-regular and even randomized/Monte-Carlo variants (see Rapidly exploring random tree). Space-filling trees have interesting parallels in nature, including fluid distribution systems, vascular networks, and fractal plant growth, and many interesting connections to L-systems in computer science.

A space-filling tree is defined by an iterative process whereby a single point in a continuous space is connected via a continuous path to every other point in the space by a path of finite length, and for every point in the space, there is at least one path that converges to it.

The concept of a "space-filling tree" in this sense was described in Chapter 15 of Mandelbrot's influential book The Fractal Geometry of Nature (1982).^[4] The concept was made more rigorous and given the name "space-filling tree" in a 2009 tech report ^[5] that defines "space-filling" and "tree" differently than their traditional definitions in mathematics. As explained in the space-filling curve article, in 1890, Peano found the first space-filling curve, and by Jordan's 1887 definition, which is now standard, a curve is a single function, not a sequence of functions. The curve is "space filling" because it is "a curve whose range contains the entire 2-dimensional unit square" (as explained in the first sentence of space-filling curve).

In contrast, a space-filling tree, as defined in the tech report, is not a single tree. It is only a sequence of trees. The paper says "A space-filling tree is actually defined as an infinite sequence of trees". It defines ${\displaystyle T_{\text{square}}}$ as a "sequence of trees", then states "${\displaystyle T_{\text{square}}}$ is a space-filling tree". It is not space-filling in the standard sense of including the entire 2-dimensional unit square. Instead, the paper defines it as having trees in the sequence coming arbitrarily close to every point. It states "A tree sequence T is called 'space filling' in a space X if for every x ∈ X, there exists a path in the tree that starts at the root and converges to x.". The standard term for this concept is that it includes a set of points that is dense everywhere in the unit square.

The simplest example of a space-filling tree is one that fills a square planar region. The images illustrate the construction for the planar region ${\displaystyle [0,1]^{2}\subset \mathbb {R} ^{2}}$. At each iteration, additional branches are added to the existing trees.

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  Square space-filling tree (Iteration 1)
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  Square space-filling tree (Iteration 2)
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  Square space-filling tree (Iteration 3)
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  Square space-filling tree (Iteration 4)
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  Square space-filling tree (Iteration 5)
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  Square space-filling tree (Iteration 6)

Space-filling trees can also be defined for a variety of other shapes and volumes. Below is the subdivision scheme used to define a space-filling for a triangular region. At each iteration, additional branches are added to the existing trees connecting the center of each triangle to the centers of the four subtriangles.

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  Subdivision scheme for the first three iterations of the triangle space-filling tree

The first six iterations of the triangle space-filling tree are illustrated below:

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  Triangle space-filling tree (Iteration 1)
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  Triangle space-filling tree (Iteration 2)
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  Triangle space-filling tree (Iteration 3)
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  Triangle space-filling tree (Iteration 4)
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  Triangle space-filling tree (Iteration 5)
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  Triangle space-filling tree (Iteration 6)

Space-filling trees can also be constructed in higher dimensions. The simplest examples are cubes in ${\displaystyle \mathbb {R} ^{3}}$ and hypercubes in ${\displaystyle \mathbb {R} ^{n}}$. A similar sequence of iterations used for the square space-filling tree can be used for hypercubes. The third iteration of such a space-filling tree in ${\displaystyle \mathbb {R} ^{3}}$ is illustrated below:

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  Cube space-filling tree (Iteration 3)

• H tree
• Space-filling curve
• Rapidly exploring random tree (RRTs)
• Binary space partitioning