May 13, 2021

Trees by Jakob Steiner | Science

Trees by Jakob Steiner | Science


As we saw last week, with two points only one elementary tree can be formed on one side, and with three points the solution is also unique, while with four points there are two possibilities. With five points there are three different possibilities, and six with six points, and with seven points you can form eleven different trees, which gives rise to the sequence: 1, 1, 2, 3, 6, 11 …

What are the following numbers? Is there a formula that gives us the nth term based on n? How is this "arboreal" sequence different from the famous and ubiquitous Fibonacci succession?

In nature there are many examples of arborescent structures, such as hydrographic networks, in which the nodes would be the sources of the different water currents and the confluence points of two or more of them. Obviously, in nature, nodes are not points; but the network can be schematized in the form of an arboreal graph. And the same can be said of our circulatory system and our nervous system. Or the electrical and hydraulic networks that supply our cities.

In every tree of n nodes there are n-1 edges. It is easy to see if we imagine the trees formed by n-1 nodes "with tail" (that we can visualize as matches, and even use them to compose trees on a table) and one without.

Within the arboreal graphs binary trees have special relevance, which are those in which each branch bifurcates into two others. An illustrious example is found in the Tree of Porphyry, mentioned last week: it is a philosophical tree that goes from the general to the particular and in which each concept is subdivided into two. Thus, the being can be (worth the redundancy) corporeal or incorporeal; the corporeal can be animate or inanimate; the animate can be sensitive or insensitive …

Minimum trees

Given a set of elements, it may be of special interest to connect them in tree optimally, in the sense of achieving the shortest global interconnection, that is, the one in which the sum of all the sides is minimal; and that interconnection of minimum length is a tree of Steiner, named in honor of the Swiss mathematician Jakob Steiner, one of the greatest geometers of the nineteenth century. With the particularity that to obtain a minimum tree can be added to the initial set some points -called Steiner points- that give rise to shorter edges.

Trees of Jakob Steiner

A simple example is that of the vertices of an equilateral triangle, A, B and C. An obvious way to connect them is to draw two of the sides, for example, AB and BC. But if we take the center of the triangle ABC, S, as Steiner's point, and join it with the three initial points, we obtain a tree of lesser length than the sum of two sides. What is the overall length of this tree? Is it minimal? What would the Steiner tree be like from the vertices of an unequal triangle? And the one of the vertices of a square?

Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them Damn physics, Damn mathematics or The big game. He was a screenwriter The Cristal ball.

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