1. Introduction
Tree biomass estimates influence our understanding of both carbon sequestration [
1,
2] and fuels management [
3]. As the behavior of wildfire is strongly influenced by biomass pools in the forest [
4,
5], treatments designed to reduce biomass, have become common in California and throughout the western United States. The United States Forest Service has treated about 10 million hectares over the last 10 years. With the increasing emphasis on fire resilient systems [
6,
7,
8], foresters need to accurately quantify effects of management on aboveground biomass.
Therefore, not only are the accurate equations needed to quantify the past biomass or carbon storage [
1], but we also need to know whether these equations are affected by management activities or other factors that may vary temporally or spatially [
9].
Estimates of aboveground biomass often rely on allometric equations for individual trees with coefficients varying by species. These functions may reflect total aboveground biomass, or some component thereof. Bole volume is important because a large proportion of biomass is found in the main stem. Nonetheless, branch and foliage biomass are important from the standpoint of fire behavior [
9]. Crown biomass contributes to estimates of canopy bulk density. Through litterfall, tree crowns are a source for small material (<7.62 cm in diameter) on the forest floor. Biomass of the small branches and leaves both distributed in the crowns and on the forest floor, have great influence on rate of spread and the ability for fire to spread into the crown.
Ponderosa pine (
Pinus ponderosa Lawson & C. Lawson) is a commercially valuable tree species in North America and is a common species within much of California’s 7.7 million ha of coniferous forests [
10]. Yet there is little published information on aboveground biomass for this species [
11]. Three published sources for ponderosa pine include a sample of 9 trees from Fort Valley Experimental Forest in the southwest [
12], a sample of 23 trees in central Oregon [
13], and a sample of 21 trees in the inland northwest [
14,
15].
These equations are often simple expressions relating tree-level biomass to some expression of tree size [
12,
16]. Commonly, the independent variable is breast height diameter [
16], although some have also used crown dimensions or stand density [
13,
14,
17].
Biomass and carbon may be estimated with equations matched to species and geographic region of interest. If such equations are not available however, any equation for the species may be used [
10]. In some instances, species may be pooled [
16]. If models are not properly matched to conditions and species, estimation errors will be generated.
Of particular concern for ponderosa pine is the potential for density management to influence the dynamics of biomass and its distribution. For example, an increase of foliage growth may follow mechanical treatment (thinning) with the duration depending on thinning intensity [
18,
19]. Reliable estimates of canopy bulk density and litter fall depend on the accurate estimate of crown biomass [
7,
20]. Including crown dimensions in a biomass model will allow biomass estimation to respond to changes in crown length resulting from density management but it will not account for any differentiation resulting from trees maintaining more years of foliage or by accruing more leaf biomass per branch per year. If ponderosa pine trees can effectively increase foliage biomass quickly in response to density management, one would expect to find variation in foliage biomass relationships and perhaps branch biomass among stands with different treatment histories.
As an example of thinning effects on surface fuel accumulation through needle fall in California pine forests, trees thinned to 15 m
2 ha
−1 produced 2000 kg ha
−1 while stands at 38 m
2 ha
−1 produced 3650 kg ha
−1 after four years [
21]. Thus the rate per unit of basal area was actually greater (133
vs. 96 kg m
−2) among thinned stands of ponderosa pine.
The purpose of this study is to develop allometric relationships for trees across an array of conditions in ponderosa pine stands in northeastern California to ascertain if stand variability within the forest is a contributing factor in aboveground biomass estimation. In addition we graphically compared published equations to those fitted to evaluate the site-specific applicability of published equations.
2. Methods
The study was conducted at Blacks Mountain Experimental Forest (BMEF) in northeast California. The Experimental Forest (40°40' N, 121°10' W) is located approximately 35 km northeast of Mount Lassen and elevation ranges from 1700 to 2100 m. Soils are classed as Typic Argixerolls with mesic soil temperatures at lower elevations. Annual precipitation averages 460 mm falling primarily as snow between October and May. Stands at Blacks Mountain are dominated by ponderosa pine with occasional white fir (
Abies concolor (Gord. & Glend.) Lindl. ex Hildebr.) and incense-cedar (
Calocedrus decurrens (Torr.) Florin) at higher elevations in the forest. At lower elevations, Jeffrey pine (
Pinus jeffreyi Balf.) may occasionally be found. The 3715 ha forest has a wide range of stand densities resulting from past research and management activities, as well as some recent disturbance events [
22]. Yet, about 1000 ha remain of dense even-aged stands with an age of about 100 years. Productivity on the forest is relatively uniform, with site index [
23] ranging from 20 to 27 m at a base age of 100 [
24].
We sampled stands from three separate thinning treatments within Blacks Mountain. These stands thinned from below two (T2), eight (T8), and ten (T10) years prior to sampling for biomass. In addition, trees were sampled from five different unthinned (U) areas of Blacks Mountain. A total of 82 trees with targeted ages 120 years or younger were initially sampled, and of these, three were subsequently discarded due to exhibition of characteristics of old-growth trees, and ages exceeding 135 years. All stands are similar in respect to species composition and primary age cohort (approximately 100 years). All the thinnings at BMEF were designed to remove trees from smaller diameter classes, while also favoring the retention of pine and minimizing ladder fuels and creating a uniform open canopy condition.
Trees were selected to give an approximately equal representation across a range of diameters from 12 to 52 cm. The means of sampled trees varied from 30.1 to 33.3 cm (
Table 1). Heights of sampled trees from thinned stands tended to be shorter for a given diameter, a trait that typically reflects the increased taper associated with lower stand densities. Because of potential conflicts with an existing experiment, sample size was restricted in T
10.
Sampled trees were felled and sectioned in 2007 during late spring and summer. Disks were removed from the bole at the stump (0.3 m), breast height (1.37 m), half way to the base of the live crown, and then at five equally spaced points within the crown, beginning with the base of the live crown. The dimensions (diameter of two axes inside and outside bark and disk thickness at the end of each axis) of each sampled disk and length of each section were recorded in the field.
Table 1.
Mean breast height (1.37 m) diameter (dbh), height (h), crown ratio (cr), pre-treatment basal area (BAp) and current basal area (BAc) for sampled biomass trees in four different stands (U, T2, T8, T10) at Blacks Mountain Experimental Forest.
Table 1.
Mean breast height (1.37 m) diameter (dbh), height (h), crown ratio (cr), pre-treatment basal area (BAp) and current basal area (BAc) for sampled biomass trees in four different stands (U, T2, T8, T10) at Blacks Mountain Experimental Forest.
Treatment | Sample Size | dbh (cm) | Height (m) | cr | BAp (m2 ha−1) | BAc (m2 ha−1) |
---|
U | 21 | 32.8 | 17.3 | 0.54 | – | 66 |
T2 | 29 | 33.3 | 15.2 | 0.59 | 66 | 16 |
T8 | 21 | 33.3 | 14.8 | 0.62 | 64 | 17 |
T10 | 8 | 30.1 | 14.8 | 0.61 | 42 | 18 |
The crown was divided into five equal sections for branch sampling and diameter measurement (
Figure 1). Basal diameter of all live branches was measured within each of the five sections. Branches were selected for weight from each section in the following manner: one representative branch with average basal diameter from Sections 1, 4 and 5 (numbering from the base of the crown) and two from Sections 2 and 3. The top of the main stem, as far as needles were retained (3 to 9 years of growth), was removed. For each selected branch and the top of the main stem, foliage was stripped and weighed separately from the wood.
Figure 1.
Trees crowns were broken into five sections and portions of the bole were also removed at the stump (0.3 m), breast height and mid-crown.
Figure 1.
Trees crowns were broken into five sections and portions of the bole were also removed at the stump (0.3 m), breast height and mid-crown.
The samples for bole wood, branches and foliage were all weighed separately after being oven-dried at 80 °C for two weeks to ensure a stable weight. Total bole biomass was estimated for each tree section from the average of density at either end of each section and the dimensions of that section.
Equations for branch wood biomass (bw) and branch foliage biomass (fw) were developed as a function of branch basal diameter, section and years after thinning. Current year’s foliage was separated to obtain start-of-growing-season foliage biomass. In addition, for each sampled branch, the terminal foliage (starting at the tip of the branch and working back as far as green needles are retained) was separated for each year of production and recorded as weights indexed by years 0 through 9. Year 0 weight was recorded as zero for samples obtained after the end of the growing season. Year 1 biomass was always recorded as the most recent complete year’s foliage.
The number of years of foliage retained per branch was evaluated for each section of the tree and each of the four thinning treatments. We evaluated years of foliage retention in a linear model as a function of section identifier.
Branch and bole weights in this analysis all include bark. The foliage weight analyzed excluded the current growing season’s needle production and thus represented weight at the start of the growing season. Branch diameters ranged from 7 to 83 mm, oven-dry foliage weight ranged from 5 to 3322 g (
Table 2).
Table 2.
Minimum, mean and maximum for branch diameter (mm) and weight (g) by crown position for weight-sampled branches.
Table 2.
Minimum, mean and maximum for branch diameter (mm) and weight (g) by crown position for weight-sampled branches.
Section | Branch | Wood | Foliage |
---|
Diameter | Weight | Weight |
---|
| min. | mean | max. | min. | mean | max. | min. | mean | max. |
1 | 13 | 41 | 70 | 33 | 1122 | 4241 | 16 | 403 | 1425 |
2 | 8 | 36 | 83 | 4 | 1186 | 7477 | 5 | 437 | 3322 |
3 | 7 | 34 | 80 | 3 | 891 | 6379 | 5 | 388 | 2507 |
4 | 11 | 32 | 53 | 21 | 417 | 1536 | 18 | 281 | 964 |
5 | 9 | 23 | 39 | 6 | 116 | 489 | 11 | 135 | 412 |
Wood biomass (g) divided by volume (cm3), was obtained for each weighed bole section and then averaged over the volume between the sections. Multiplying this value by the volume of each bole section yielded biomass per section. Total oven-dry bole biomass was derived by summation over all sections.
Initial graphical analysis of trends among the trees from untreated areas of the forest indicated no trends across different untreated stands at Blacks Mountain. This was consistent with our observation of fairly uniform conditions within these areas across the entire Experimental Forest. For this reason, in all subsequent analyses we aggregated the untreated trees together into one treatment group (U).
2.1. Branch Wood and Foliage Biomass
The individual branch foliage biomass was fit to data pooled within a stand using weighted nonlinear regression:
where
fw is foliage biomass (g),
kit is model parameter
i at treatment
t,
bdob is branch basal diameter outside bark (mm), and
εfw is a random branch-level error with assumed variance structure

.
The individual branch woody biomass was fit using weighted nonlinear regression:
where
bw is branch woody biomass (g),
qit is model parameter
i at treatment
t,
cp is branch crown position (ranging from 0.1 to 0.9) using the center point of the section’s position within the crown and
εbw is random branch-level error with

.
These fitted regressions were then applied to each branch to obtain wood and foliage biomass for each branch and, through branch summation [
15], the entire tree.
2.2. Tree Biomass
Using unconstrained iterative seemingly unrelated regression (SUR) [
25,
26,
27], we fit a system of equations with three response variables: foliage, branch-wood and bole-wood biomass. The simultaneous fit allowed for improvements in efficiency due to cross-equation correlations in the errors of the system. A full model was specified as:
In this model, a, b, and c are unknown parameters, the response variable is tree component biomass (g), It is a treatment area indicator (0,1) specified so that, for example, the parameter a00 is associated with the unthinned trees and a01 is associated with trees in a stand observed two years after thinning, etc., cr is observed crown ratio (live crown length divided by total height), dbh is breast height diameter (cm) and height is total tree height (cm).
Heteroskedasticity for the three components of the model was addressed by transformation:
Various combinations of a reduced form were fit by removing stand-identifier terms (indicator variables) and crown ratio from the model. A final model was selected after fitting the full model and fourteen selected reduced forms of the system. Model selection was guided by an information-criteria approach [
28]. Employing an assumption of multivariate normality, Akaike’s information criterion (
AIC) was calculated for each fit [
29], with a small sample correction (
AICc) then applied [
30]:
where the matrix:

,
N is sample size,
p is number of models in the system, and
K is the number of system parameters estimated (including covariance terms).
2.3. Foliage Production
We estimated the annual allocation of foliage by proportioning annual foliage for the terminal in each sampled branch. We then fit the current full year’s foliage as a function of the estimated total for the terminal. This gives a proportional estimator that was used to estimate the amount of current year’s foliage in any given tree. We tried models with slope adjustments for thinning, for section, for both section and thinning, and with no adjustments. We employed a nonlinear, mixed-model with a random tree-level effect (ε
s) for the slope term to obtain the following predictive model for current annual branch foliage (
af):
where
rit are parameter estimates for treatment areas
t = 1, 2, 3, 4,
af is the last full year’s branch foliage biomass (g) and

is the predicted branch total from model 1. Indicator variables were used for sections and treatment:
i1t = 1 for Sections 1–3, 0 otherwise;
i2t = 1 for Section 4, 0 otherwise;
i3t = 1 for Section 5, 0 otherwise. The branch-level error term assumed to have variance proportional to prediction,

. This model was then used to estimate the amount of current annual foliage production for each tree by summation over all branches.
We graphically evaluated the number of years of needle retention for branches by treatment area and position in the tree. We then fit a tree level (linear) response for annual foliage biomass for individual trees to evaluate changes in litter fall within the Experimental Forest.
2.4. Application
To evaluate the stand level impact of thinning on total foliage biomass and production rates, we applied the fitted model to a thinned stand sampled in this study (T2) at Blacks Mountain Experimental Forest. The selected stand for this application was thinned in 2005 and then sampled five years post-thinning. Biomass was estimated pre- and post-thin from a sample of 45 variable radius points with a basal area factor of 5.6 m2 ha−1. Prior to thinning, this was a dense second growth stand with approximately 781 trees ha−1 and a basal area of 53.5 m2 ha−1 (S.E. = 3.0), the quadratic mean diameter was 18.8 cm. After thinning from below (i.e., removal of trees from lower crown classes to favor trees in the larger crown classes), density was approximately 112 trees ha-1 and basal area was reduced to 10.6 m2 ha−1 (S.E. = 1.0). This thinning from below elevated quadratic mean diameter to 34.9 cm. Five years post-thinning, basal area had increased to 11.9 m2 ha−1 (S.E. = 1.0). We obtained pre-thin and five-year post-thin estimates of both the branch, foliage biomass and the annual foliage production. We also estimated five-year post-thin biomass using the unthinned model to evaluate the stand-level error generated by mis-application of the model.
At the tree-level, we graphically evaluated predictions from three published biomass models [
12,
13,
14] for foliage, branches and boles with the models fitted for BMEF, to ascertain differences across the range of observed diameters.