Branching Structures

We can use a classification system to assign a numerical value to each knot and to each edge of the topological branching structure, called its reference or ordinal number. In the botanical sense, a closely approximated allocation is the time of origin of the knot and/or the corresponding bud and the branch. The trunk receives the ordinal number zero, all main branches receive the ordinal number one, and all branches receive higher order numbers.

The number of branches of an order corresponds then to the shoots grown in one year, and the relationship between this number and the number of branches

of the underlying order is described by the branching ratio. In Fig. 2.3 such Section 3.2

ordinal numbers for the branches of a model are indicated. Branching Structures

In this way, to each tree statistical values may be assigned that describe how many shoots per year or per branching order are generated and also how the branching takes place. In Sect. 4.8 an approach is described that, through the direct simulation of such statistical values, renders trees that are astonishingly real looking.

The Horton-Strahler analysis, a method commonly used for the analysis of ^ Horton-Strahler analysis

branching structures in geology and hydrology, differs from the botanically

motivated analysis [95,210]. While in the latter case, evolutionary history plays

a significant role, the Horton-Strahler analysis provides a growth-independent

measure. Here, the Strahler number of the root is determined, which is also

called the Strahler order. Strahler order

Подпись: ord(vm) Подпись: max(ord(vi),ord(vj)) if ord(vi) = ord(vj) ord(vi) + 1 else Подпись: (3.1)

All terminal nodes are assigned the order one. For the inner nodes vm with children Vi and vj this results in the order number ord(vm):

Подпись: (a) Branching Structures Подпись: Figure 3.2 (a) Tree with order numbers and Strahler numbers; (b) perfect tree; (c) degenerated thin tree
Branching Structures

While the perfect tree in Fig. 3.2b receives the Strahler number four, the order of the degenerated tree is two. A segment of the order к is the maximum result of connected edges with the order number к. It follows that the tree in Fig. 3.2a has two segments of order two.

Outside of botany, large branching structures are especially found in hydrol­ogy. Several studies discuss the analysis of river networks. Here especially the ^ river networks bifurcation ratios, f3k, play a role. They show how frequently the segments of order к branch. If bk is the number of segments with order к, fdk = bk-1/bk is valid, and for the tree in Fig. 3.2a the bifurcation ratio results are в2 = 8/2 = 4 and вз = 2/1 = 2.

The minimal number for a pi is two, and is obtained for all orders with the perfect binary tree in Fig. 3.2b, while the degenerated tree can, relative to its length, reach an arbitrary bifurcation ratio.

In Sect. 4.9 we introduce a process that is an extension of the Horton-Strahler analysis for the production of branching structures with a combinatory ap­proach. In this approach, the order is expanded to a biorder that includes the directed branching. This allows the description of the branching characteristics

of a tree using a so-called branching matrix, which then is used for generating the geometry.