FBA: Fractal Branch Analysis model

Introduction: trees of many kinds

Trees come in various shapes, grow at different rates and interact with their neighbours during development. Yet, many of the properties of an individual tree can be predicted if we know the diameter of its stem. The relationship between this diameter and properties such as tree height, tree biomass, leaf area and harvestable timber are called ‘scaling rules' or allometrics. Empirical allometric scaling equations (the most generic form is Y = a D^b) for tree biomass Y on the basis of stem diameter D are often used in forest inventories and assessment of carbon and nutrient stocks in vegetation; they are based on cutting selected trees and obtaining destructive measurements to relate to the stem diameter. When shifting from plantation forestry to mixed forestry or multi-species agroforestry systems, however, short-cuts to the empirical approach are desirable. Certain regularities in the development of tree form are captured in ‘fractal branching' models; such models can provide a transparent scheme for deriving tree-specific scaling rules on the basis of easily observable, non-destructive methods. Apart from total tree biomass, the models can provide rules for total leaf area, relative allocation of current growth to leaves, branches, stem or litter, or the ratio of green to brown projection area that modulates tree-crop interactions in savanna.

An example of tree shapes obtained by varying just 1 parameter in the fractal branching routine: the proportionality factor p for change of stem diameter at a branching point has the values 0.8, 1.0, 1.2, and 1.4 respectively in figures A-D. Trees with low p value are endowed with more branches and leaves; those with high p have less branches and leaves, due to more significant branch tapering.

Belowground, similar descriptions hold for individual root axes, where the proximal root diameter at the stem base can be used for predicting total length or biomass of all its branches. The basic assumptions underlying fractal branching have been tested and found to be applicable as acceptable approximation for a wide range of tropical trees, aboveground as well as for their root systems.

An example of tree root architecture produced by the FBA model with top (A) and lateral (B) view.

The functional branch analysis protocol and program are designed to efficiently describe the architecture and key properties of a tree, and to use the derived parameters to reconstruct trees with simple, repetitive (‘fractal') rules and derive scaling rules that relate stem and/or proximal root diameter to total biomass and other properties.