Modelling vertical allocation of tree stem and branch volume for hardwoods
Modelling vertical allocation of tree stem and branch volume for hardwoods
Neil R. Ver Planck 0
David W. MacFarlane 0
0 Department of Forestry, Michigan State University , 480 Wilson Road Room 126, East Lansing, MI 48824-1222 , USA
Whole-tree volume equations are in great demand due to the need to quantify the distribution of wood volume within trees for estimating whole-tree utilization potential. While main stem volume has been extensively studied, related to computing the merchantable timber volume of trees, the relative volume of branches has received much less attention. It is particularly challenging to quantify branch volume in trees with deliquescent branching architecture (i.e. hardwoods) where branching is complex and not strongly controlled by a dominant stem. Here, new mixed-effects cumulative volume profiles are presented that allow for simultaneous volume estimation of the dominant stem and whole tree from ground to the top of the tree. Cumulative branch volume can be estimated at different relative heights from the whole-tree and dominant stem profiles by simple subtraction. The models were developed from destructive sampling of 32 trees from a temperate hardwood forest in Michigan, US. The species in the sample were primarily American beech (Fagus grandifolia Ehrh.) and sugar maple (Acer saccharum Marsh.). The results produce whole-tree cumulative volume models that include all branches in trees and demonstrates the value of studying the whole tree even when the dominant stem is the object of interest.
Introduction
Models for predicting components of tree volume have been
a hallmark of forest measurement science since its
inception, beginning with simple volume tables to estimate bole
saw timber volume to a fixed merchantable top diameter
(Gevorkiantz and Olsen, 1955), progressing to increasingly more
sophisticated stem profile models, aka ‘taper’ models, which can
predict the volume of the main stem to any height from base to
tip (Jordan et al., 2005). Clearly, the next phase in tree volume
modelling must be to describe the ‘whole’ tree, i.e. predicting at
least tree branch volumes, if not the volume of the root systems,
along with the volume of the main stem. Whole-tree volume
models are needed to inform sustainable usage of whole trees
(Flewelling, 2004; Zakrzewski, 2011), not just their merchantable
boles. Tree volume estimation should also be made
compatible with whole-tree biomass estimation and carbon accounting
systems (MacFarlane, 2011; Van Deusen and Roesch, 2011).
Although whole-tree biomass models are widely
available, whole-tree volume models are lacking. Only a few
approaches have been developed specifically for estimating
whole-tree, above-ground volume including a centroid-based
method (MacFarlane, 2010), a volume expansion factor method
(MacFarlane, 2011) and an estimation method based on
importance sampling (Van Deusen and Roesch, 2011). Recently,
MacFarlane (2010) and Zakrzewski (2011) published cumulative
volume profiles compatible with taper modelling theory that
included wood from both the main stem and large
(merchantable) branches, but these works did not include smaller
(non-merchantable) branches.
Here, new whole-tree volume profile models are presented,
including volume from all branches in trees, with the goal
of advancing the field of whole-tree modelling, while
accommodating the continued need for accurate and flexible
estimation of merchantable main stem volume. A segmented
modelling approach is used for fitting of two separate
models (for segments below and above crown height) into a
whole-tree model similar to the segmented approach used
by Max and Burkhart (1976). The crown height is fixed as
the join point for the two models, similar to the switching
model of Valentine and Gregoire (2001), respecting the onset
of branching as a critical inflection point in tree stem form
(Adu-Bredu et al., 2008; MacFarlane, 2010). A mixed-effects
modelling approach is also employed to reflect the within-tree
correlations (Valentine and Gregoire, 2001; Leites and Robinson,
2004; Westfall and Scott, 2010) and because mixed-effects
modelling often performs better than a fixed-effects modelling
approach (Valentine and Gregoire, 2001).
The manuscript is organized as follows. First, a whole-tree
model is described along with associated whole-tree and
dominant stem cumulative volume profile models. Then, the models
Figure 1 A simplified hardwood tree schematic showing the location of
branch volume accumulation along the dominant stem as a function of
relative height, ranging from zero at base to one at top of the tree. The
first branches encountered are at relative crown height (rC).
are fit to real trees and the profiles are examined to look at
concurrent shifts in whole-tree vs dominant stem volume profiles,
including shifts in the centroids of volume.
Whole-tree and dominant stem models
We began with a model of a whole tree that contains a ‘dominant’
stem to which a se (...truncated)