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Hyperbolic tree

Hyperbolic tree is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hyperbolic tree rather than just read about it. In short: A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry. Displaying hierarchical data as a tree suffers from visual clutter as the number of nodes per level can grow exponentially.

Hyperbolic tree — main illustration
Hyperbolic tree — illustration

Key takeaways

  • Hyperbolic tree belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Hyperbolic tree to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Hyperbolic tree from memory before moving on to harder problems.

Reference excerpt

A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry.

Displaying hierarchical data as a tree suffers from visual clutter as the number of nodes per level can grow exponentially. For a simple binary tree, the maximum number of nodes at a level n is 2n, while the number of nodes for trees with more branching grows much more quickly. Drawing the tree as a node-link diagram thus requires exponential amounts of space to be displayed. One approach is to use a hyperbolic tree, first introduced by Lamping et al. Hyperbolic trees employ hyperbolic space, which intrinsically has "more room" than Euclidean space. For instance, linearly increasing the radius of a circle in Euclidean space increases its circumference linearly, while the same circle in hyperbolic space would have its circumference increase exponentially. Exploiting this property allows laying out the tree in hyperbolic space in an uncluttered manner: placing a node far enough from its parent gives the node almost the same amount of space as its parent for laying out its own children. Displaying a hyperbolic tree commonly utilizes the Poincaré disk model of hyperbolic geometry, though the Klein-Beltrami model can also be used. Both display the entire hyperbolic plane within a unit disk, making the entire tree visible at once. The unit disk gives a fish-eye lens view of the plane, giving more emphasis to nodes which are in focus and displaying nodes further out of focus closer to the boundary of the disk. Traversing the hyperbolic tree requires Möbius transformations of the space, bringing new nodes into focus and moving higher levels of the hierarchy out of view. Hyperbolic trees were patented in the U.S. by Xerox in 1996, but the patent has since expired.

See also Hyperbolic geometry Binary tiling Information visualization Radial tree – is also circular, but uses linear geometry. Tree (data structure) Tree (graph theory)

References

External links

d3-hypertree – HTML5 Hyperbolic tree implementation, MIT licensed Hyperbolic Tree of life – Open source tree of life visualisation using Open Tree of Life data set The Green Tree of Life – Tree of life – University of California at Berkeley and Jepson Herbaria Tree of life Similar to the above, but with pictures RogueViz supports hyperbolic trees.

Illustrations

Hyperbolic tree: A basic hyperbolic tree. Nodes in focus are placed in the center and given more room, while out-of-focus nodes are compressed near the boundaries.
A basic hyperbolic tree. Nodes in focus are placed in the center and given more room, while out-of-focus nodes are compressed near the boundaries.
Hyperbolic tree: Focusing on a different node brings it and its children to the center of the disk, while uninteresting portions of the tree are compressed.
Focusing on a different node brings it and its children to the center of the disk, while uninteresting portions of the tree are compressed.

Worked examples

Example 1 — a first encounter with Hyperbolic tree

Start with the simplest possible case. Write down what Hyperbolic tree claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hyperbolic tree before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hyperbolic tree ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hyperbolic tree

In research
Hyperbolic tree appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Hyperbolic tree in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Hyperbolic tree is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph drawing, Hyperbolic geometry, Trees (data structures), so understanding it makes those chapters shorter.
In everyday life
Look for Hyperbolic tree outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hyperbolic tree in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Hyperbolic tree means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Hyperbolic tree out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hyperbolic tree in simple terms?

A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry. Displaying hierarchical data as a tree suffers from visual clutter as the number of nodes per level can grow exponentially.

Why does Hyperbolic tree matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hyperbolic tree?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hyperbolic tree.

Tags

  • Graph drawing
  • Hyperbolic geometry
  • Trees (data structures)

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