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.

(A)	(B)

(C)	(D)

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.

fba1

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.

How does it work?

Fractal branching models
repeatedly apply the same equations to derive subsequent orders of the
branching process (‘self-repetition rule'). For practical applications, a rule
is added for stopping when a certain minimum size is reached. The rules can
refer to the diameter, length and/or orientation of the next order of branches.
A figure below describes the elements of a 'functional branching analysis'
(FBA) scheme, which can be applied for above as well as belowground parts of
trees. The combinations of the various parameters can be used to predict total
size (weight, surface area, length, height, lateral extent) and the allometric
scaling equations between these.

Elements of the Functional Branch Analysis (FBA) model
to derive allometric scaling equations between above or belowground tree parts

fba1

Estimation of the model's elements is based on a
visual assessment or field observation. The model needs information of link
(i.e. shoot or root segment) diameter and length, and final structure (leaves
or fine roots). Not all, but at least 50 and preferably 100 successive links
are to be measured to get a precise estimate of branch parameters. The model's elements
governing branching pattern can be calculated with FBA Help-File and the
independency of p (proportionality factor) and q (equity factor) to link
diameter should be checked since it (i.e. the independency) is an underlying
requisite to apply the self-repetition rule.

How good is it?

One
comparison between model estimation and real observation on tree aboveground
biomass and its part was done related to four tropical trees grow in the Philippines and
shown in the figure below. Total aboveground tree biomass as calculated with
the allometric equations from FBA model, fits well with the biomass
measurements obtained from destructive methods (A). Slight differences were
found for tree components: wood (B) and leave biomass (C) for all four tree
spices. Statistical test analysis also confirms the viability of the FBA model
for all tree species. Indeed, all test performed on FBA results indicated that
the model is applicable and provides an acceptable approximation for total
aboveground biomass estimation as well as for the tree components (wood and
leaf).

Comparison between FBA estimation and direct harvest
biomass values for four tropical tree species in the Philippines (Martin, 2008): A. tree
aboveground biomass (kg), B. wood biomass (kg) and C. leaves biomass (kg).

fba1

Field data collection

Basically
the model needs three kinds of information to estimate tree biomass:
information of tree size and branching pattern, information of woody components
and information of the final structure of the tree.

Tree size and branching pattern

Field
data of link (i.e. branch or root segment) diameter and length have to be collected
and the diameter is ideally measured twice, cross-wise, at three positions on each
link (describe in the figure below): proximal, middle, and distal. Link number
and its parent number should be recorded as follow: the main stem is given link
number 1, its offspring are number 2 and 3 (i.e. if it has two offspring). The
parent number of main stem is zero (described in the table below). The number
of leaves (single or compound leaves) of each link should be also recorded. For
reliable estimates of the fractal branching parameters, minimum number of 100
branching points should be collected for each tree sample.

Positions to measure diameter of each link (Dproximal,
Dmiddle, and Ddistal). The diameter measurement should be done perpendicular to
branch angle. At each position, the diameter is ideally measured twice, cross
wise, to consider a non-circular branch shape

fba1

Field data required to measure tree size and branching
pattern for the FBA model. D_prox, D_mid, and D_dist
are the average diameter at proximal, middle, and distal position of each link
respectively

Link no	Link length	D_prox	D_dist	D_mid	Parent no	N of leaves
1	L₁	D_p1	D_d1	D_m1	0	n₁
2	L₂	D_p2	D_d2	D_m2	1	n₂
3	L₃	D_p3	D_d3	D_m3	1	n₃
4	L₄	D_p4	D_d4	D_m4	2	n₄
...

Woody components

The model classifies the
woody part of the tree into three categories: wood, branch and twig. The classification
follows the diameter of the link. The dry weight per volume of wood (wood
density) should be measured for each of the three woody components.

Final structure of the tree

The length of bare tip on
final links (see Figure 2), the average surface area (one side only) of a
single leaf, and specific leaf area (SLA) defined as the surface area of leaves
per unit dry weight (cm²/g) are to be measured. For belowground, the
required parameters are the average of fine root (usually defined as those with
D < 2 cm) length and specific root length (SRL) defined as fine root length
per unit fine root dry weight (cm/g).

References

Martin, F. S. 2008. Using native timber trees for
recovering degraded landscapes in the Philippines: social, biophysical
and economic assessment of agroforestry systems practiced by smallholder
farmers. Doctoral Thesis, Cordoba University,
Madrid, Spain.
149 pp.

Van Noordwijk, M., Lusiana, B., and Khasanah, M. 2004.
WaNuLCAS version 3.1, Background on a model of water, nutrient, and light
capture in agroforestry systems. International Centre for Research in
Agroforestry (ICRAF), Bogor,
Indonesia. 246
pp.

Van Noordwijk, M. and
Mulia, R., 2002. Functional branch
analysis as tool for fractal scaling above- and belowground trees for their
additive and non-additive properties. Ecol. Model. 149, 41-51.

Van Noordwijk, M. and Purnomosidhi, P. 1995. Root
architecture in relation to tree-soil-crop interactions and shoot pruning in
agroforestry. Agrofor. Syst. 30, 161-173.

Van Noordwijk, M., Spek,
L. Y., and De Willigen, P. 1994. Proximal
root diameter as predictor of total root size for fractal branching models I.
Theory. Plant Soil 164, 107-117.