L -systems are extremely simple algorithms: visualize a turtle moving a distance d straight forward. At the end of its journey it produces two offspring, one branching of slightly to the left and the second branching off slightly to the right, and both traveling a 10% shorter distance. Once the offspring reach their destination, each creates two offspring which branch off slightly to the left and slightly to the right and move straight ahead a distance which is only 10% shorter than that of their parents. This process is repeated for seven generations. Figure 1 shows the trails of the turtles. The trails form a ramified structure, a turtle graph, which resembles the branching structure of trees and weeds. Figure 2 is an example for the branching pattern of a weed. Compared to the branching patterns of plants, turtle graphs appear too regular. However if the length of the trails of the offspring is randomized by 50% and some turtles produce only one offspring or none, then the turtle graphs a surprisingly similar to plant structures. Figure 1 contains a turtle graph where offspring are produced with a 90% probability. To illustrate that the code is very short and simple, we show a MatLab TM version in Table 1.The parameters of turtle graphs can be used to match the appearance of plants. The branching angle and the ratio of the trail length of the offspring and the parent are the genetic properties of plant species. The survival rate of the offspring depends on the general well-being of the plant determined by the climate and soil conditions. Weather, deseases, and other short term fluctuations influence the magnitude of the fluctuations of the trail length.The mathematics of deterministic turtle graphs is amazingly simple. The number of endpoints L 5 2 g , where g is the number of generations and the number of branch points is Y 5 2 g 21. For turtle graphs where the ratio of the trail length of offspring and parents r < 1, the distance from the start to each endpoint is R 5 d (1 2 r g )/(12r), never exceeds the value R 5 d/(12r) even for an unlimited number of generations. The height and the width have limiting values too. This means that the width and height of the plant never exceeds these limits even if the plant has an unlimited life span. However, little is known about the expectation values of the height and width and other quantities of probabilistic turtle graphs.The technical term for turtle graph is L-system, named after the Hungarian biologist Aristid Lindenmayer [1] or iterated function system [2]. They have been used to model plants for animated movies [3] and to analyze and model other complex data, such as music [4]. Hornby [5] uses graphs to model the modularity
ALFRED HÜ BLERAlfred Hü bler is the Director of the