Periodic forests of stunted trees

Abstract

We may define a forest of stunted trees as follows: Consider an infinite background of nodes at the vertices of an infinite plane tessellation of equilateral triangles., and start from a straight line of nodes at unit distance apart, which we shall consider as the ground ; other parallel lines of nodes are then spaced at successive levels of linearly increasing heights above the ground. Any node may be live (if a tree passes through it) or vacant otherwise. Any live node may give rise to a branch to one or other or both of the two nearest nodes at the next higher level, but this growth is stunted , on either side, if the neighbouring node on that side is also live and could provide a branch to the same higher level node (this other branch is also stunted). Many of the figures in the paper show the type of forest that results. The Introduction, § 1, describes the origin of this idea, and §2 gives definitions and points out certain basic properties and ideas for combining forests and for separating them into simpler units. A variety of periodicities is discussed. In §3 a mathematical theory is developed in terms of generating functions expressed as power series. Sequences and forests are represented by ratios ⌀(t)/f(t) of polynomials with coefficients in GF (2). A matrix formulation is also defined. The theory is developed in §4, so that periods and forests can be developed from those for basic sets having irreducible polynomials f(t) as denominators, with co-prime numerators of lower degree. In §5, the determination of base- and row-periods for particular irreducible polynomials f(t) is investigated as a preliminary to the enumeration of forests with given base-period n in §6, and of reflexive forests in §7. Further interesting properties, problems and applications are discussed in §8 ; it is intended to develop some of these in another paper. The tables give enumerations and properties connected with sequences and forests generated by various polynomials f(t) of low degree, culminating in table 5, which gives the numbers of forests with base periods up to 50, and table 6, which lists all individual forests with n up to 15. M any of these forests are given in the diagrams, intended to bring out various symmetry properties and possible variations.