# From Dendrochronology to Allometry

## Abstract

**:**

## 1. Introduction

## 2. What Is Wrong with This Picture?

_{t}, mm) of each tree ring is equal to the difference between the current stem radius (R

_{t}, cm) and the prior year one (R

_{t−1}, cm). Ring area (a

_{t}, mm

^{2}) is the difference between the area of a circle with radius R

_{t}and the area of a smaller circle with radius R

_{t−1}. When a ring is locally absent, it is given a zero value for width or area, rather than considering it as a missing observation. The number of years from the pith is called cambial age, a value that refers to a tree ring, not to the vascular cambium, whose physiological ability to divide and form new tissue defies the commonly held notions of aging and senescence [9]. Basal area, i.e., the transverse surface of the trunk, is typically calculated from stem diameter at ‘breast height’ (1.3–1.5 m from the ground) measured outside bark. Diameter is converted to area assuming a perfect circle, and radial growth, or basal area increment (BAI), is computed as the difference between consecutive basal areas. Annual BAI is equivalent to ring area, which however can refer to any position along the stem, not just breast height. Ring area is also independent of bark thickness, which changes between species as well as with tree age and size, stem position, and site conditions [10].

_{t}− w

_{t−}

_{1}), represents the rate of change in growth increment with respect to time, which is an index of growth acceleration [11]. Authors who favor measures of growth acceleration usually apply a logarithmic transformation beforehand to stabilize the variance of the time series [16], assuming that the relative, rather than the absolute, growth fluctuation is affected by an underlying trend [17] (p. 94). Differencing tree-ring widths or areas results in frequency distributions no longer bound by a minimum value of zero. It also generates values for locally absent rings that are not immediately identifiable because they correspond to growth accelerations with varying non-positive values.

_{t}), which is the increment with respect to initial plant size, and relative growth change (RGC

_{t}), which is the increase in size with respect to the previous increment. The following section uses simple equations to clarify the abovementioned concepts, mostly derived from the forestry and plant population biology literature, in terms of dendrochronological measurements.

## 3. Simple Mathematical Relationships

_{t}) could be expressed as:

_{t}). When growth acceleration is computed from log-transformed increments (∆l

_{t}), it becomes a log-transformed relative growth change. This statement can be demonstrated using ring widths, as follows (here the positive constant k needs to first be added to avoid taking the logarithm of zero):

_{t}) is defined for each year t as the relative difference between two consecutive averages of 10-year ring-width periods, one that ends at year t and another that immediately follows. In mathematical notation, it can be written as:

## 4. A Visual Example of Tree-Ring Data

## 5. From Dendrochronology to Allometry

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Graphical representation of tree-ring sequences from seven cross sections taken at multiple heights along a conifer trunk. BH = breast height (~1.3–1.5 m from the ground). There are multiple errors in this cartoon, which provide a learning opportunity and are discussed in the text.

**Figure 2.**The stem cross-section known as MLK, an acronym derived from the collector’s (Earl H. Morris) last name initial and the field site (Lukachukai Mountains, within the Navajo Nation in northeast Arizona, USA). It is a Douglas-fir (or PSME, shorthand for Pseudotsuga menziesii (Mirb.) Franco) specimen uncovered from the archaeological excavation of Broken Flute Cave (photo © Laboratory of Tree-Ring Research, Tucson, Arizona; courtesy of Rex Adams). Being an archaeological specimen, there is no bark left on the outside; only the wood remains. The shape of the rings is nearly circular, with yellow-colored earlywood and brown-colored latewood forming each annual growth increment. Year 550 is clearly indicated by an arrow pointing at the conventional two-dot symbol, which is used to mark the half-century year. Other visible single-dot symbols mark each decade, and a three-dot symbol marks the century, year 600 in this case. By a twist of chance, calendar decades on this specimen happen to correspond to cambial decades, so for instance year 550 occurs at the cambial age of 30 years. A change in overall color occurs around year 580, from darker (inside; ~55–60 rings) to lighter (outside; ~45–40 rings). Within the overall trend from wider inner rings to narrower outer ones, there are easy-to-spot ‘pointer’ years [34], i.e., 526, 536, and 543. One of these relatively sudden, single-year growth decreases, year 536, is in common with many other trees and sites, possibly because of a global climatic anomaly caused by one or more major volcanic eruptions [35] or by a comet impact [36]. As one would expect, these hypotheses are quite contentious.

**Figure 3.**Time series plots of ring-width measurements and computed growth values for the MLK cross-section (Figure 2). The horizontal x-axis, which represents time, has the same length and scale in every plot. The vertical y-axis, which represents growth, has the same length in every plot and covers the range of the plotted variable, hence the vertical scale is not constant among plots. The growth measurement unit is in parentheses except for RGCw and RGCa, which are dimensionless ratios. For ∆lw and ∆la the measurement unit is in brackets because of the need to add a unit constant to avoid taking the logarithm of zero. A vertical dashed red line was drawn to connect values for the same cambial age, and to visualize the one-year shift of the reversed peak generated by using either relative growth changes or first differences compared to ring widths or ring areas. Cumulative growth curves were also plotted from ring widths (radius) and ring areas (basal area). RGCw = relative growth change (width); RGCa = relative growth change (area); ∆w = first difference of ring widths = growth (radial increment) acceleration; ∆a = first difference of ring areas = growth (basal area increment) acceleration; ∆lw = first difference of log-transformed ring widths = log-transformed relative growth change (using radial increment); ∆la = first difference of log-transformed ring areas = log-transformed relative growth change (using basal area increment); RGw = relative growth (width); RGa = relative growth (area); RGR = relative growth rate; GC = percentage growth change (from means); RMED = percentage growth change (from medians).

**Figure 4.**A revised version of Figure 1, without dating errors. The ring sequences in each section were redrawn to better illustrate the concept of crossdating. Color lines are used to identify the three-dimensional ordering of growth layers within a tree. The cambial age of a ring is defined as the number of years from the pith. There are three different ordering schemes, or sequences: (a) transverse, or type 1 sequence, where rings of increasing cambial age are formed over time at the same stem height (cross sections BH and 1 through 6); (b) tangential, or type 2 sequence, where rings of the same cambial age are formed over time at increasing stem heights (red lines connect the rings included in one such sequence); (c) oblique, or type 3 sequence, where rings of increasing cambial age are formed in the same year at increasing stem heights (green lines connect the rings that form one of these sequences). These three ordering schemes correspond to different ways in which tree rings are influenced by environmental variability and internal physiological processes [49] (p. 9).

**Figure 5.**A correct way to represent both stem analysis and crossdating. The transverse sections are lined up vertically according to calendar years, which were assigned to the rings by visually crossdating their common patterns across samples. Notice that the ring-width sequences are the same as those in Figure 4. Without properly assigning dates to all growth layers, errors can be made in the analysis of tree growth, either for an individual stem or for a forest, and in any calculations related to changes over time in diameter, height, taper, volume, biomass, and carbon content. Lack of temporal control prevents accurate identification of factors that drive wood formation, thus crossdating becomes crucial for any type of tree growth study at interannual and longer time scales.

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Biondi, F.
From Dendrochronology to Allometry. *Forests* **2020**, *11*, 146.
https://doi.org/10.3390/f11020146

**AMA Style**

Biondi F.
From Dendrochronology to Allometry. *Forests*. 2020; 11(2):146.
https://doi.org/10.3390/f11020146

**Chicago/Turabian Style**

Biondi, Franco.
2020. "From Dendrochronology to Allometry" *Forests* 11, no. 2: 146.
https://doi.org/10.3390/f11020146