A Brief History of Strahler Numbers

The Strahler number or Horton-Strahler number of a tree, originally introduced in geophysics, has a surprisingly rich theory. We sketch some milestones in its history, and its connection to arithmetic expressions, graph traversing, decision problems for context-free languages, Parikh’s theorem, and Newton’s procedure for approximating zeros of differentiable functions. 1 The Strahler Number In 1945, the geophysicist Robert Horton found it useful to associate a stream order to a system of rivers (geophysicists seem to prefer the term ‘stream”) [20]. Unbranched fingertip tributaries are always designated as of order 1, tributaries or streams of the 2d order receive branches or tributaries of the 1st order, but these only; a 3d order stream must receive one or more tributaries of the 2d order but may also receive 1st order tributaries. A 4th order stream receives branches of the 3d and usually also of lower orders, and so on. Several years later, Arthur N. Strahler replaced this ambiguous definition by a simpler one, very easy to compute [26]: The smallest, or ”finger-tip”, channels constitute the first-order segments. [. . . ]. A second-order segment is formed by the junction of any two first-order streams; a third-order segment is formed by the joining of any two second order streams, etc. Streams of lower order joining a higher order stream do not change the order of the higher stream. Thus, if a first-order stream joins a second-order stream, it remains a second-order stream. Figure 1 shows the Strahler number for a fragment of the course of the Elbe river with some of its tributaries. The stream system is of order 4. From a computer science point of view, stream systems are just trees. Definition 1. Let t be a tree with root r. The Strahler number of t, denoted by S(t), is inductively defined as follows. – If r has no children (i.e., t has only one node), then S(t) = 0.