Allometric relationships between sapwood area and shrub dimensions for six common Southern African savanna bush encroacher species: Universal or species‐specific?

Southern African savanna rangelands are facing a widespread degradation pattern called bush encroachment. This is associated with implications for various aspects of the water cycle and in particular canopy transpiration. At the individual‐tree scale, it is estimated by scaling sap‐flux density by sapwood area. However, the direct measurement of sapwood area is impracticable at landscape scale and general allometric equations of the West‐Brown‐Enquist (WBE) model relating sapwood area to primary size measures seem to fail for some species and climates. Therefore, we conducted intensive field measurements to establish species‐specific allometric relationships between sapwood area and sizes (stem diameter, crown area) in six dominant shrub species involved in bush encroachment in Namibia (Colophospermum mopane, Senegalia mellifera, Vachellia reficiens, Dichrostachys cinerea, Vachellia nebrownii, Catophractes alexandri). We found strong allometric relationships between sapwood area and stem diameter as well as between sapwood area and crown area for all six species. These relations are largely in line with the WBE theory but still provide estimates that are more accurate. Only in D. cinerea, the sapwood area was significantly smaller than predicted by the WBE theory, which might be caused by a larger need for stabilizing heartwood. Our results are useful to estimate water loss via transpiration at a large scale using remote sensing techniques and can promote our understanding of the ecohydrological conditions that drive species‐specific bush encroachment in savannas. This is particularly important in the light of climate change, which is considered to have major implications on ecohydrological processes in savannas.

High temperatures and solar radiation make evapotranspiration the major form of water loss, which accounts for water loss amounting to over 90% of the annual precipitation (Haan et al., 1994).It includes water evaporating from the soil and from the plant foliage after being intercepted as well as water transpired by plants following soil water uptake.Thus, even relatively small changes of vegetation composition and structure could have important consequences, not only for soil moisture budgets but also for groundwater recharge, land-atmosphere energy exchange, local climate and primary production including carbon accumulation (Huxman et al., 2005;Lubczynski, 2009).Hence, estimating evapotranspirative water loss is critical for managing the environmental effects of bush encroachment.A common constraint is, however, the difficulty in obtaining evapotranspiration estimates and its partitioning across widespread and sometimes inaccessible savannas.This difficulty is often closely associated with speciesspecific traits of water use of the woody species, which are involved in a particular bush encroachment process (Lubczynski et al., 2017).
It has become increasingly apparent that the estimation of evapotranspiration depends on sufficient understanding and accurate modelling, including validation, of the two main processes underlying evapotranspiration, namely, transpiration and evaporation.While estimating evaporation is relatively straightforward, determining the rate of vegetation water use is far more complex (Sun et al., 2019).
Transpiration is influenced by many factors including species-specific rooting depth, conductive sapwood area and canopy structure (Sohel, 2022).
A common method to estimate transpiration of woody vegetation relies on measurements of sap-flux density (SFD) across the active fluid-transporting tissue (the xylem), also known as sapwood.Multiplication of SFD with sapwood area provides the total water-use of an individual tree, while the multiplication with total sapwood area of all woody plants per ground area provides an estimate measure of woody stand transpiration at landscape scale.In general, the per-species SFD variability among trees of different size and age is relatively low (Jaskierniak et al., 2016;Kumagai et al., 2007;Reyes-Acosta & Lubczynski, 2014).Therefore, tree water use depends mainly on the conductive sapwood area.A reliable estimate of sapwood area, therefore, is a key component in quantifying transpiration of the woody vegetation components in bush encroached savannas.
However, measuring sapwood area is time consuming, often destructive and impractical, both at the small scale of a plot or stand and at landscape level.This is particularly true for shrub stands occurring in semi-arid savannas, which often comprise of multi-stemmed species.While some variability of sapwood patterns among species exists (Carrodus, 1972), fortunately, sapwood area scales reliably with various plant dimensions, such as leaf area, stem diameter and crown dimensions, which are easier to assess (Lubczynski et al., 2017;Mitra et al., 2020;West et al., 1999).Particularly, crown properties can later be detected and quantified at landscape scale using aerial imagery, light detection and ranging (LiDAR).Although previous attempts to establish allometric relationships between sapwood area and crown dimensions, such as crown area or crown diameter, have found that they can in fact be strong (Fregoso, 2002;Mitra et al., 2020), stem diameter still achieved the most accurate values of sapwood area.A convenient model for explaining characteristic allometric relations of different species between specific size measures, such as stem diameter, and traits, such as sapwood area, is the West-Brown-Enquist (WBE) model of West et al. (1999, hereafter WBE).It predicts that sapwood area scales with stem diameter with a third power law of 7/3 (2.33).
Reasons for the constant scaling are geometric and hydrodynamic constraints that limit the size in living organisms (West et al., 1997).
Despite many findings supporting the WBE model's predictions, empirical results showed that variations from the predicted scaling value do occur (Daba & Soromessa, 2019;Schoppach et al., 2021).
Therefore, establishing species-specific relationships between sapwood area and stem diameter may be essential to accurately estimating transpiration rates (Ter-Mikaelian & Korzukhin, 1997;Yaemphum et al., 2022).This may be particularly important in savanna shrubs, which may increase sapwood area relative to the total wood area, avoiding cavitation-embolism under drought (Brodersen & McElrone, 2013).
Allometric equations for estimating sapwood area and eventually whole-tree and stand transpiration using tree size parameters have been developed for various woody ecosystems, such as tropical and temperate forests (Wang et al., 2010;Yaemphum et al., 2022), but only a few exist for arid and semi-arid bush encroached savannas.
Studies for Southern Africa are particularly rare (Lubczynski et al., 2017).
The primary goal of this study was therefore to test whether allometric relations between sapwood area and stem diameter or crown area exist for six dominant bush encroacher species of Namibia.Relying on extensive field measurements, we derived species-specific equations, which can later be used in combination with sap flux monitoring to estimate canopy transpiration for different bush encroachment scenarios at plot and landscape scale, with the latter based on remote sensing of crown area.Second, we tested whether the allometric relations between sapwood area and stem diameter agree with scaling predictions from the WBE model.
We predicted that deviations from the model are attributed to drought induced species-specific changes of relative investments in the area of sapwood.

| Study area
The study took place in the Etosha Heights Private Reserve (EHPR) (19.2451 S, 15.1921 E), located in the Kunene region in northern Namibia, approximately 400 km north of the city of Windhoek.EHPR spreads over 480 km 2 and shares an approximately 65 km long border with the Etosha National Park.The research site is part of a largescale wildlife area.
The climate in the region is semi-arid (Peel et al., 2007).It is characterized by high air and soil temperatures with mean annual temperatures above 22 C and high solar radiation.Annual rainfall amounts to 300-350 mm, and precipitation is highly variable in time and space.
Annual evaporation rate ranges 2500-2600 mm, an order of magnitude higher than the average levels of rainfall (Atlas of Namibia Team, 2022).Temperatures and precipitation are changing seasonally, with the highest average precipitation during the wet season (November-March) and lowest average precipitation during the dry season (April-October).The soil is rich in carbonate and consists of limestone and dolomite rocks as well as sand and calcrete (Nortjé, 2019).The vegetation type in the area is a tree-and-shrub savanna with varying density, from large open grasslands with scattered trees up to dense, shrub-dominated landscapes (Atlas of Namibia Team, 2022).The most dominant shrub species in the region is Colophospermum mopane, which occurs in high abundance throughout the reserve (Bester, 1996).Other prominent species in the area include Senegalia mellifera, Dichrostachys cinerea, Vachellia nebrownii, Catophractes alexandri and Vachellia reficiens.The fauna within EHPR is rich and includes large populations of species from various taxonomic classes, including large mammalian grazers (antelopes and giraffes), browsers (elephants and rhinos) and predators (lions and leopards).

| Shrub sampling and measurements
We sampled all prominent shrub species, in total six species, in an area of approximately 164 km 2 in central-east EHPR during March and April 2021.Sampling was assigned to 60 blocks, with a minimal distance between blocks of 500 m to control for local effects and to avoid spatial autocorrelation.Measured individuals and stems reflect the full range of sizes for each species occurring within EHPR.To avoid biases in shrub dimensions due to browsing, we sampled only individuals with minimal apparent damage, such as recently broken branches or a large portion of missing foliage caused by, for example, rhinos or elephants.
To establish allometric relationships between canopy area (CA) and sapwood area via stem diameter, we measured CA and stem diameter (D Stem ) for one individual of each of the six species (360 individuals in total, 60 per species) in each block.CA estimates relied on the average crown spread (ACS) method (Blozan, 2006), using measuring poles with a 1 cm scale.As stems are not necessarily round, D Stem was measured by averaging two perpendicular diameters using a vernier scale with an accuracy level of 1 mm.In the case of a multistemmed individual, we measured all stems.Although D Stem is commonly measured at breast height (i.e., 130 cm above ground level), the morphology of most shrub species included in this study did not allow for this measure, as shrubs are often either smaller or far too branched at this height.Since the selected species differ in size and morphology, we set appropriate heights for each species as follows: 20 cm (Catophractes alexandri, Vachellia nebrownii), 50 cm (D. cinerea, Senegalia mellifera, Vachellia reficiens) and 130 cm (C.mopane).The validity and comparability of stem diameter measurements at different heights rely on the foundations of Leonardo da Vinci's 'rule of trees', which states that the total cross-section area across all stems in an individual plant is rather constant along the height axis (Richter, 1970).The pipe model theory, which offered a mechanistic explanation for this self-similarity of stem area along the height axis in vascular plants, observes trees as systems of conducting pipes connecting roots with leaf nodes (Shinozaki et al., 1964).Since the sapwood tissue is composed of multiple continuous xylem vessels that stretch from the roots to the leaf nodes, the total sapwood area can be assumed to vary little between cross sections at different heights as well (Magalhães & Seifert, 2012;Oppelt et al., 2001).
To study the relationship between stem diameter and sapwood area, we randomly selected between 20 and 23 individuals per species (visiting 25 blocks: 0-1 individual per block per species, 130 individuals in total).We removed one branch per individual using branch shears or a handsaw.To ensure that our data set for developing the allometries contained samples from the complete spectrum of D stem values, we sampled a large range of branch sizes (3.9-111.4mm).
Measurements of stem diameter and sapwood area were achieved via staining of stem cross sections.Immediately after cutting, the ends were shaved with fresh razor blades, thus reducing embolism risk as well as other damages to the xylem or blocks that could have occurred during the harvest.The sample was immediately inserted into a tube containing a staining solution of methylene blue 0.5% (w/v) and distilled water with a concentration ratio of 1:20 (Sellin et al., 2008).The amount of diluted solution varied according to the diameter of the sample and the tubes used for staining, between 10 mL for very small samples (twigs) and 200 mL (large branches).
Tubes were sealed to prevent evaporation and samples remained in the solution for at least 24 h to allow uptake of the dye into the stem either via active transpiration by the remaining foliage or via capillary action through xylem vessels.After removing and drying the samples in the air, cross-sections were made approximately 1 cm above the bottom of each sample using branch shears or an electric hand saw.The cross-sections were placed on a clear background adjacent to a ruler with 1 mm marks for consistent scaling.Images were taken using a digital Canon SX540 HS camera placed on a tripod.Visual analysis was done using ImageJ (Fiji is Just) version 2.1.0/1.53c(Abramoff et al., 2003;Schindelin et al., 2012).
Representative examples for staining samples for each species are illustrated in Figure 1.
In our measurements, we assumed that the entire region between the heartwood and the bark constitutes sapwood.Since distinguishing bark from sapwood was consistently achievable through visual examination, staining samples primarily served to identify the innermost sapwood ring and establish the boundary between sapwood and heartwood.In every sample, the borders of the following three areas were marked manually and the resulting areas measured digitally with an accuracy of 1 mm 2 : (A) total area (the entire area of the cross-section including bark), (B) wood area and (C) heartwood area (Figure 2).The outer limits of the heartwood were defined as the first innermost tree ring of the crosssection not containing traces of blue staining (Figure 2).While our staining approach dyed the active part of the sapwood only, this procedure enabled us to identify the entire sapwood area that has been formed.Sapwood area was mathematically obtained by subtracting the heartwood area from the wood area.Stem diameter (D stem ) was derived based on total area (A tot ), assuming circularity of the cross-section.

| Data analysis
Species-specific allometric relationships were modelled using the power law equation (Niklas, 1994) where X is the size measure (D stem or CA), b a normalization constant, a the scaling exponent and Y the estimated sapwood area (SA or SA tot ).We considered the total sapwood area (SA tot ) of each study shrub for the allometry between CA and SA.To obtain SA tot , we measured the D stem of all stems, applied the corresponding speciesspecific allometric relationship between D stem to SA to each stem, and summed up the estimates.To identify the scaling exponent and to enable a comparison with the WBE prediction, we applied logtransformation and fitted a linear model as Models were assessed using both, R 2 and the regression coefficient a (the scaling exponent) and its significance (p < 0.05 of F-value).
Additionally, 95% confidence intervals of the scaling exponent were calculated and plotted for each species-specific model.
The equations were back-transformed into their 'natural' exponential form to simplify the application of the allometric relationships.
To prevent underestimation bias of the logarithmic regression estimate (Niklas, 1994), a back-correction factor CF was introduced based on Sprugel (1983) as where SEE is the standard error of the scaling exponent estimate a.
The final equation for the prediction of sapwood area from the respective size measure will then have the following form: To identify species-specific structural characteristics of sapwood formation due to drought and their relation to plant age, we also calculated the relative sapwood area ( SA wood area ) and fitted an additional linear model for the relationship between relative sapwood area and D stem for each species.

| RESULTS
In total, we stained and measured 130 individual plants for sapwood area and stem diameter, and measured CA and total stem diameter of 360 individuals of six savanna bush encroacher species.A detailed summary of the measurements is found in Table A1.
In all six species, sapwood area was positively correlated with stem diameter.D stem accounted for almost the entire variation in C. alexandri (99%), C. mopane (98%), S. mellifera (97%), V. reficiens (97%), V. nebrownii (97%) and D. cinerea (78%) (Figure 3, Table 1).The very high values of R 2 for all species established D stem as a strong independent variable to estimate sapwood area.Apart from D. cinerea, all species included the regression line predicted by the WBE model within the 95% confidence interval of the slope.
The relative sapwood area to D stem relationship varied substantially between species (Figure 4).While this ratio was constant in C. mopane, S. mellifera and V. reficiens, a significant positive relationship was found in C. alexandri, V. nebrownii and a significant negative one in D. cinerea.On average, D. cinerea invested clearly the least of all species in water conducting sapwood compared to heartwood.
F I G U R E 3 Allometric relationship between stem diameter (D stem ) and sapwood area (SA) and relationship between stem diameter (D stem ) for six main shrub encroacher species in a semi-arid savanna in Namibia.Solid black lines are the regression lines calculated by the models, dashed black lines are the 95% confidence intervals, and the grey line is the predicted relationship according to the West-Brown-Enquist (WBE) model (slope = 2.33) (West et al., 1999) with species-specific intercepts.
Although R 2 values were high for all species, they were lower than for the correlation between D stem and SA (Figure 3, Table 1).
T A B L E 1 Allometric exponents with 95% confidence interval, their test statistics, and the back-correction factor CF for the relationship between stem diameter and sapwood area for six main shrub encroacher species in a semi-arid savanna of Namibia.CF is needed for prediction of SA from SD applying back-transformation of the allometric relationship at ln-scale (see Equation 4and Figure 3).F I G U R E 4 Relative sapwood area (relative SA) for six main shrub encroacher species in a semi-arid savanna in Namibia.Relative sapwood area is sapwood area divided by the wood area of the cross-section.
F I G U R E 5 Relationship between crown area (CA) and total sapwood area (SAtot) for six main shrub encroacher species in a semi-arid savanna of Namibia.Solid lines are the regression lines calculated by the models; dashed black lines are the 95% confidence intervals.
T A B L E 2 Species-specific allometric exponents with 95% confidence interval, their test statistics, and the back-correction factor CF for the relationship between crown area and total sapwood area of six main shrub encroacher species in a semi-arid savanna of Namibia.CF is needed for prediction of total sapwood area from SAtot applying back-transformation of the allometric relationship at ln-scale (see Equation 4and Figure 5).Sapwood area is a key component in quantifying canopy transpiration, and allometric equations for estimating sapwood area using tree size parameters have been developed for various woody ecosystems.And yet, in the case of Southern African savannas, only few exist (Lubczynski et al., 2017), although various allometric relationships have been extensively studied in recent years (Fregoso, 2002;Issoufou et al., 2015;Moncrieff et al., 2011;Tredennick et al., 2013).
Scaling exponents of these size-correlated trends often rely on the universal value predicted by the WBE model (Niklas, 1994;West et al., 1999).Nevertheless, the model itself is a point of dispute (Brown et al., 2005;Kozlowski & Konarzewski, 2004) because it does not incorporate features specific for different plant taxa or for different environments.As sapwood formation is likely governed by low water availability, establishing species-specific relationships between sapwood area and tree dimension for Southern African savanna woody species may be essential to accurately estimating transpiration rates (Ter-Mikaelian & Korzukhin, 1997;Yaemphum et al., 2022).In this study, we established allometric relationships between tree dimension and sapwood area in six shrub species involved in savanna bush encroachment in Namibia.
Our results illustrate that sapwood area can be reliably predicted based on stem diameter in all six species tested.Comparing our results with the prediction of the WBE model (West et al., 1999) showed that the model holds true in most cases.
Various findings illustrate that despite the unique conditions in savannas and other drylands, the scaling relationships between stem diameter and sapwood area in drought resistant plants are similar to those of plants found in other climates (Fregoso, 2002;Gebauer et al., 2009;Patino et al., 1995;Wang et al., 2010).However, the relationship between stem diameter and sapwood area in D. cinerea revealed an exponent substantially lower than the other five bush species tested.This is not exclusive for the Etosha region.In the eastern Kalahari in Botswana, D. cinerea revealed the smallest slope of nine species (Lubczynski et al., 2017) with wider conduits or with more conduits in total.Such a sapwood composition at the expense of parenchyma is, however, more vulnerable to cavitation and embolism (Brodersen & McElrone, 2013), even if the same conductivity across the stem is reached compared to species exhibiting the allometric exponent of 2.33 of the WBE theory.Parenchyma cells in sapwood serve as large water storage reservoirs and are linked to osmotically driven embolism repair mechanism (Brodersen & McElrone, 2013).The vulnerability to cavitation and embolism might be partly compensated by a decrease in conduit size.
The higher wood density of D. cinerea compared to other savanna species (Fernández-Ortuño et al., 2015) supports indeed a smaller conduit size.The notion that these vessels are small, although probably high in number, finds additional support by findings showing that D. cinerea has a low sap flow velocity relative to other savanna shrub species (Zziwa, 2003), and estimations of transpiration rates of vegetation plots in areas predominated by D. cinerea were lower than areas with similar shrub density predominated by other common shrubs (Chavarro-Rincon, 2009).Although the size and number of conducting vessels per stem area of D. cinerea is unknown, the allometric exponent and sap flow velocity taken together indicate that the species might be relying on this to compensate for a reduced sapwood area.Thus, D. cinerea seems to be less efficient in water uptake and might be more water dependent than the other bush encroacher species in our study.Indeed, D. cinerea does almost not increase in biomass and is less abundant in areas with less soil moisture and higher temperature (de Klerk, 2004;Shikangalah et al., 2021).Its main distribution in Namibia as bush encroacher extends to savannas with MAP >550 mm (de Klerk, 2004).Questions remain regarding the underlying causes of its dominance in these moister areas as well as the functional driver of the unusual small allometric exponent.In any case, D. cinerea's striking anatomy is supported by our results on low relative sapwood area and by previous results showing that D. cinerea has an unusual small sapwood to heartwood ratio than other woody savanna species (Shikangalah et al., 2021;Zziwa, 2003).A large heartwood area might relate to an adaptation mechanism, which increases the stem stability, because heartwood is generally stiffer than sapwood.A strong argument for this view is our observation that relative sapwood area decreased with age (stem diameter as a proxy of age or growth).However, if central heartwood was to increase flexural stiffness, the heartwood would need to have an inordinately lower elasticity than sapwood (Niklas, 1997), a trait which we do not know.
Moreover, high wind velocities as a driver of stability traits are not typical for the main distribution areas of D. cinerea (Atlas of Namibia Team, 2022).
An adaptation in flexural stiffness or any other reason for relatively more heartwood was not detected in C. mopane, S. mellifera and V. reficiens.Furthermore, the small shrubs V. nebrownii and C. alexandri even decrease the relative amount of central heartwood with stem age, promoting a higher flexibility instead of stiffness, which might be affordable if the increase in overall woody biomass contributes to stability already.If sapwood area is correlated with storage capacity for water (Scholz et al., 2008), lower water availability could be a driver distinguishing their local occurrence.While definitive conclusions regarding C. mopane, S. mellifera and V. reficiens based on these findings are currently hard to make, our results could help understand what conditions drive which shrub species during bush encroachment.
Based on the overall strong relationship between stem diameter and sapwood area, we were also able to establish strong relationships between crown area and sapwood area in all six species.This result is meaningful because it supplies further support for the potential of estimating sapwood area based on aerial imagery (Mitra et al., 2020).
Whereas previous studies found a linear relationship between crown size and properties of water conductance (Ahongshangbam et al., 2020;Quiñonez-Piñ on & Valeo, 2019;Tziaferidis et al., 2021), the relationships detected in our study were slightly exponential, presumably due to the inclusion of smaller samples.Another reason for a divergent allometric relationship might be the time of sampling.We carried out data collection and measurements at the end of the rainy season.Deciduous phenology of savanna trees and shrubs has been shown to be strongly influenced by seasonality (Dahlin et al., 2016), meaning that the relationships between crown area and total sapwood area may vary seasonally.Continuous measurements of crown area and stem diameter as a base for total sapwood area are therefore necessary to adapt the relationship described above to different seasons and years.
In addition, since our measurements of plant dimensions were

| CONCLUSIONS
The assessments of six main bush encroacher species of Namibia proposed here indicate that sapwood area can be reliably predicted based on stem diameter, crown dimensions and the universal WBE model.
Our allometric equations offer the prospect of vastly increasing our knowledge about transpirational water losses in semi-arid bushencroached savannas.Robust species-specific models predicting individual shrub sapwood area from shrub size are, however, critical for accurate estimating of stand-and landscape-scale transpiration in shrub encroached regions, in particular when D. cinerea is involved.A subsequent extension of our research may apply these speciesspecific total sapwood area-crown area or sapwood area-stem diameter allometric equations to estimate woody transpiration for different bush encroachment scenarios in the studied region.Such analysis remains highly relevant for all Southern African savannas given the widespread shift in vegetation composition via bush encroachment by different species but also considering climate change, in particular as droughts are projected to become longer and more frequent (IPBES, 2018;IPCC, 2023).

F
I G U R E 2 Delineation of sapwood area based on sapwood staining in Catophractes alexandri.The following areas are outlined by three white lines: (a) total area, (b) wood area and (c) heartwood area.
taken within a nature reserve, where browsing pressure by large African mammals such as giraffes, elephants and rhinos is present, a comparison of crown area across different land-use types with different browsing pressures might be required to understand to what degree crown area is affected by the presence of large browsers.Nevertheless, we believe that our allometric relations are sufficiently reliable and universal due to our selection of apparently unharmed individuals, as well as due to the sampling of shrubs from multiple plots covering a large area with many different local conditions.