Ecological Archives M085-009-A1

Stephen C. Sillett, Robert Van Pelt, Allyson L. Carroll, Russell D. Kramer, Anthony R. Ambrose, and D’Arcy Trask. 2015. How do tree structure and old age affect growth potential of California redwoods? Ecological Monographs 85:180–211.

Appendix A. Supplemental methods used to quantify tree structure, age, and growth.

Additional trees

In addition to the 13 old-growth forests where we installed 1-ha plots, we measured 14 SEGI from Whitaker Forest, one SEGI from Kings Canyon National Park, one SESE in Redwood Experimental Forest, and two SESE on private land between 2005–2013 using the same methods. To be comprehensive, we used ground-level LiDAR combined with other measurements to reconstruct the main trunk and crown of one more tree (SEGI 43), the only individual larger than every tree in our dataset (Van Pelt 2001) and for which climbing and direct measurement were not permitted.

We positioned a C-10 HDS Scanner (Leica Geosystems) at 18 ground-level stations around one tree (SEGI 43) between 5 and 300 m away from the trunk to minimize obstructions, maximize field of view, and produce a three-dimensional point cloud of the tree. Angle deflection at each station was 360° horizontal and 270° vertical (+90 to –45°), and scan point density was set between 1 and 10 cm at 100 m. Eighteen reflective targets were positioned around the tree to provide a geometric basis for combining data from multiple scans into a single point cloud with < 1 cm resolution. After isolating the tree's points from those of ground and other vegetation, we rotated data using Cyclone (version 7.3.3, Leica Geosystems) to display all points in the horizontal plane and depict a transverse section of the crown. We computed crown spread as the functional diameter of a polygon created by tracing the crown perimeter in Photoshop. Crown depth was measured as the vertical distance between crown base and highest leaf, which was clearly evident when the point cloud was displayed in the vertical plane. Just as with other large, crown-mapped SEGI, we calculated crown volume using crown spread, crown depth, and the formula for a prolate spheroid. After isolating the main trunk's points from those of appendages, we generated cross-sections of the trunk at average ground level, high point of ground (0.8 m), 1.4 m, and then 1-m height intervals up to 62 m. Above this height, there were too few points to derive an accurate trunk perimeter, so we used previously published ground-based laser measurements of diameter to reconstruct the main trunk, which terminated in a long-dead spire (Van Pelt 2001). Cross-sections above 15 m that included buttresses beneath appendages were ignored to replicate what we would have obtained by wrapping a tape around more or less circular portions of the trunk. To simulate diameter measurement with a tape, we used ImageJ to create a convex hull of each polygon made by tracing the trunk perimeter in Photoshop. Cross-sections at 15 m and below were converted to polygons (ignoring bark fissures) to replicate those obtained for trunks of all other trees. Polygon cross-sections were then converted to radii for use in main trunk calculations.

Measurement details

Crown mapping—A plumb line, compass, and steel tape were used to quantify trunk lean (if any). Distances and azimuths were measured with steel tape or survey laser and compass. Ends of segments, termed nodes, were numbered so that each segment had a unique name defined by its proximal and distal nodes. Branches were numbered sequentially and recorded for node of origin. Branch slope was measured with a clinometer from branch base to centroid of foliage. Branches arising mid-segment received additional measurements of distance from nearest node and clock position (around circumference) of origin. Non-round segments (e.g., buttressed limbs) were treated as ellipses; a steel tape was used to measure vertical and horizontal diameters, and the average was used to avoid overestimation.

Branch sampling—Most SEGI, but not SESE, branch samples had ovulate cones, which were separated into green and brown categories during dissection. All cones were counted, oven-dried, and weighed, revealing substantial branch-level variation in cone size. Prior to drying, a subsample of 55 green cones from 8 branches with varying cone size was measured for length and width (excluding peduncle) to the nearest 0.1 mm to calculate surface area using the equation for a prolate spheroid. A linear equation was then used to convert dry mass (g) to fresh area (cm²) for green cones (mass = 1.1046 × area + 7.6042, R² = 0.96) for each branch.

Substantial within-crown variation in ovulate cone abundance necessitated a different quantitative approach than other SEGI branch components. During crown mapping, a total of 89995 cones were counted on 1896 randomly selected foliar units. Most of a tree's cones occurred in the upper third of the crown, so equations predicting numbers of green and brown cones per foliar unit used height or relative height. Foliar units on each tree were tallied for upper, middle, and lower thirds of the crown. Mean number of green and brown cones per foliar unit was predicted for each portion of the crown. To account for tree-level variation in cone abundance, actual cone counts per foliar unit were regressed against predicted values for each tree, and slope of this linear relationship was used as a scalar to adjust predicted values.

Small branches quantities of both species were computed via multiples rather than allometric equations, because these branches were counted rather than mapped. For SESE, midpoint diameters for the 0–2 and 2–4 cm diameter size classes were used to predict components for each size class using equations for branches < 8 cm diameter. However, analysis of dissected stems revealed that branch taper behaved as a fractal (i.e., accelerating increase of path length for decreasing diameter follows a power function), causing significant deviations from conic formulae when diameter < 4 cm (R. Van Pelt, unpublished data). So revised midpoint diameters (0.933 and 2.802 cm) of the two size classes were used to derive multiples for small SESEbranches. For SEGI, components of branches < 7 cm diameter on main trunks and segments of SEGI, which were tallied as foliar units, were predicted using averages derived from 41 dissected foliar units.

For SESE, separate equations were developed for small branches using only samples < 8 cm diameter (N = 70) and large branches using only samples > 4 cm diameter (N = 140), because inclusion of large branches in the same regression yielded less reliable predictions for small branches. Equations for small branches were used to predict quantities for branches < 6 cm diameter, whereas equations for large branches were used to predict quantities for branches ≥ 6 cm diameter. For both size classes, the best predictor was base diameter, but live path length, horizontal extension, height, and relative height were also important for some quantities (Appendix B). For SEGI, the best predictor was base diameter, but live path length, # foliar units, and height were also important for some quantities (Appendix B).

Dead volume on living branches was estimated via a 3-step procedure. First, live + dead branch volume was calculated via the equation for a conic frustum using measured path length (live + dead), basal diameter, and a distal diameter of 4 cm (SESE) or 7 cm (SEGI). Second, dead volume was computed by subtracting live volume (calculated using live path length, base diameter, and distal diameter) from live + dead volume. Third, dead proportions of bark and sapwood volume, which were visually estimated during crown mapping, were added to the dead volume calculated in the second step. Volumes of completely dead branches were calculated using one of two equations. If a dead branch's calculated total length using Cartesian coordinates of its base and top was greater than measured path length, volume was calculated via the equation for a conic frustum using total length, basal diameter, and distal diameter. Otherwise, dead path length, basal diameter, and distal diameter were used.

After measuring path lengths and radii, subsamples of known lengths from each branch diameter category were oven-dried at 101°C to stable mass and weighed to the nearest 0.1 g. Average densities were used to convert fresh volumes to dry masses for all sampled branches of both species (SESE = 479 ± 2 kg m-3, N = 913;SEGI = 465 ± 4 kg m-3, N = 271; values given as mean ± one standard error hereafter). Branch wood and bark densities were computed separately by stripping subsamples of bark after drying and then reweighing (SESE wood = 490 ± 5 and bark = 350 ± 7 kg/m3, N = 75; SEGIwood = 488 ± 21 and bark = 389 ± 14 kg/m3, N = 33).

Trunk sampling—To improve quantification of SEGIbark below 5 m, which was up to 58 cm thick, 360 additional bark thickness measurements were collected on a wide variety of tree sizes and ages. In all cases, we measured bark thickness from a convex hull (as a tape wrapped around trunk) and not inside fissures. Bark densities could not be determined from increment cores as bark was compressed and often lost during extraction. To quantify trunk bark densities, discrete patches of bark were cut from freshly fallen trunks of widely varying diameters (10–400 cm) using a handsaw, hammer, and chisel. Bark volume was measured as length × width × thickness, where thickness was distance from cambium to outermost surface (i.e., location of hypothetical tape wrap). Samples, including fragments, were then oven-dried at 101°C to stable mass and weighed to the nearest 0.1 g. Average densities were used to convert fresh bark volumes to dry masses for all trunks of both species (SESE = 234 ± 9 kg/m3, N = 11;SEGI = 208 ± 6 kg/m3, N = 6).

A subset of trunk increment cores were separated into sapwood and heartwood sections and sealed in airtight tubes immediately after extraction. Wood volumes were determined by the Archimedes principle—after suspending fresh cores from an apparatus on a digital balance and submerging them in a container of distilled water standing on the bench, core volume was measured as the mass decrease to the nearest 0.0001 g. Cores were then oven-dried to stable mass and weighed to the nearest 0.0001 g. Average densities were used to convert fresh wood volumes to dry masses for trunks of both species (SESE sapwood = 358 ± 4 kg/m3, N = 254;SESE heartwood = 407 ± 6 kg/m3, N = 242; SEGI sapwood = 285 ± 6 kg m-3, N = 69;SEGI heartwood = 378 ± 11 kg/m3, N = 69).

Modeling ages

Trunk ring width series were divided into sections representing early, middle, and late periods of trunk growth history as long as each period had at least 50 annua rings. If fewer rings were available, series were divided into two sections or considered as a whole. Slow-growing suppressed or recovering individuals with low aboveground vigor (see Dimensions of tree structure) were not used to develop power functions used to predict trunk ages beyond the earliest cross-dated year. Most of these trees (6 SESE, 2 SEGI) were small enough for cores to reach pith, in which case power functions were unnecessary to determine trunk ages. We developed separate power functions for two populations in one SESE location (RNP), because annual rings were considerably larger for a given trunk size in lowland compared to upland forests (Appendix E). To develop power functions for one SEGI location (RMG), we supplemented sample size with three trees from another location (WF) within the same grove 2 km distant. We estimated ages of trees that were not cored by applying power functions to predicted wood radii of their main trunks at 10-m height intervals. In these cases, main trunk wood radius was computed as a function of total radius and relative height for SESE and as the difference between total radius and predicted bark radius for SEGI (Appendix C).

We used two methods to determine ages of appendages. First, the largest branch or limb arising from each tree's main trunk was assumed to be original, having arisen from an axillary bud just beneath the trunk's apical meristem (i.e., branch pith nearly reaches trunk pith). Epicormic appendages were excluded because they arise from the vascular cambium and cannot be aged without coring. Base height of the appendage was then used to calculate its age by interpolation of crossdated or predicted main trunk ages. Second, basal cross-sections from 153 destructively sampled branches were polished, digitally scanned, and subjected to annual ring counts along three radii from pith to cambium (i.e., 45° from top on either side and 180° from top of branch). An attempt was made to crossdate each branch using standard and novel techniques (Kramer et al. 2014), but when this was not possible (N = 115), ring count of the oldest radius from each cross-section was used to estimate age.


A combination of statistical approaches was employed in this study to capitalize on their complementary strengths (Mundry 2011). We used an information-theoretic (IT) approach to evaluate competing models and to select the best model (or combination of models) approximating the relative influence of tree structure and age on growth. When our objective was to develop allometric equations, we examined scatterplots of rigorously measured predictands (e.g., leaf area) against easily measured predictors (e.g., diameter), used power-function transformations to linearize bivariate relationships, applied stepwise regression to identify worthwhile combinations of predictors, and then manually checked various combinations for goodness of fit such that final equations—excluding those yielding negative predictions of physical quantities (e.g., biomass < 0) for small samples—had a limited number of statistically significant parameters. We also used a frequentist approach for multivariate and non-parametric analyses lacking an accessible IT analog as well as to evaluate significance of growth trends.

Statistical model fitting and AICc analyses proceeded in R (R Foundation for Statistical Computing, Vienna, Austria). Saturated models were screened for systematic lack of fit and leverage, and goodness of fit was examined with residual plots and generalized R². If selection of the best model was ambiguous (i.e., best model had < 90% of Akaike weight), we examined evidence for each parameter by calculating the AICc-weight-averaged model from the top models. Model averaging inflated standard errors on parameters to produce an unconditional standard error that incorporated uncertainty in both parameter estimation and model selection (Burnham and Anderson 2002).

Principal components analysis (PCA, McCune and Mefford 2011) was performed on the primary matrix for each species using a correlation cross-products matrix. Ovulate cones only occurred on trees south of 40° latitude, so no variables describing cone abundance were included in the primary matrix for SESE (Table 4). To obey the assumption of multivariate normality, variables exhibiting excessive skewness, kurtosis, or bivariate nonlinearity with other variables were power-transformed prior to analysis. Skewness averaged 0.18 and 0.30 and kurtosis averaged 0.23 and 0.01 in the final matrices for SESEand SEGI, respectively, and there was no pronounced bivariate nonlinearity. We used randomization tests to identify which principal components were statistically significant (Peres-Neto et al. 2005).

In addition to PCA, we subjected primary matrices of tree-level variables to two multivariate hypothesis tests (McCune and Mefford 2011). In RDA, a distance biplot was used for axis scaling, response variables were not standardized prior to analysis, and a randomization test (104 runs) was performed to determine statistical significance. In MRPP, which is a non-parametric test, no transformations were applied to variables prior to analysis.

Computer models

All Cartesian coordinates and radii generated in Microsoft Excel from field measurements were converted to a format accepted by AutoCAD (Autodesk Inc.) using a script written in Matlab (The Math Works, Inc.). AutoCAD models were exported to Acrobat Pro XI (Adobe Systems Inc.) in a universal three-dimensional (3D) format (.iges) via the Tetra4D (Bend OR) plugin. We studied tree models in the resulting 3D PDFs to locate and correct errors such as discontinuities, reverse azimuths, and abnormal diameters.

Literature cited

Burnham, K. P., and D. R. Anderson. 2002. Model selection and multi- model inference: a practical information-theoretic approach, Second edition. Springer, New York, New York, USA.

McCune, B., and M. J. Mefford. 2011. PC-ORD, multivariate analysis of ecological data, version 6.08. MjM Software, Gleneden Beach, Oregon, USA.

Mundry, R. 2011. Issues in information theory-based statistical inference—a commentary from a frequentist's perspective. Behavioral Ecology and Sociobiology 65:57–68.

Peres-Neto, P. R., D. A. Jackson, and K. M. Somers. 2005. How many principal components? Stopping rules for determining the number of non-trivial axes revisited. Computational Statistics and Data Analysis 49:974–997.

Van Pelt, R. 2001. Forest giants of the Pacific coast. University of Washington Press, Seattle, Washington, USA.

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