Modelling height–diameter relationships in living Araucaria (Araucariaceae) trees to reconstruct ancient araucarian conifer height

To reconstruct a fossil forest in three dimensions, an accurate estimation of tree height is crucial. However, modelling the height–diameter relationship of ancient trees is difficult, because the trunks of fossil trees are usually fragmentary, making direct height measurements impossible. One practical approach for reconstructing ancient tree height is to use growth models based on the height–diameter relationships of the nearest living relatives of fossil taxa. Here we apply 19 models to describe height–diameter relationships of living Araucaria trees for establishing appropriate models for ancient Araucariaceae trees. Data come from four living populations of Araucaria: A. bidwillii and A. cunninghamii in Queensland, Australia, and A. cunninghamii and A. hunsteinii in New Guinea. According to an AIC‐based model selection, a power function with an exponent of 0.67 (termed here the modified Mosbrugger model) is found to be the most appropriate for each population and for the entire dataset (157 trees), but normalization constants differ across populations. To find the most appropriate models for the genus Araucaria, 100 random samples (each population generating 25 random samples) from the entire dataset are used. Based on 100 curve fitting results on each model and multiple performance criteria, three median models are generated from the medians of their parameter estimates. Of these, the median 2pPower model works best for Araucaria, but the modified Mosbrugger and Curtis models perform nearly as well. In a case study, we revise tree heights of Upper Jurassic araucariaceous logs in Utah, USA, by applying these three models.

F O R E S T S have played a dominant role in terrestrial ecosystems on earth for more than 300 million years (Berry & Marshall 2015).Some of the most informative sources of evidence for ancient forests are preserved in the geological record as fossil stumps in original growth position or as fossil logs.Fossil forests with in-situ stumps preserve detailed information about tree density, tree age, height distribution and biomass (e.g.Mosbrugger et al. 1994), while fossil log assemblages offer data on tree age and height distribution (e.g.Gee et al. 2019Gee et al. , 2023)).For instance, a silicified log flora described from the Upper Jurassic Morrison Formation at Rainbow Draw area in Utah, USA, contains variously sized tree trunks ranging from 71 to 127 cm in maximum preserved diameter, which were used as the basis for estimating the height of ancient trees.The largest log was reconstructed as a tree with a minimum height of 28 m, and the log assemblage was interpreted as forming a forest of large conifer trees that attained individual ages of over 100 years (Gee et al. 2019).
In present-day habitats, using the measurement of diameter at breast height (DBH) to estimate tree height has been implemented in management practice and silvicultural research, because DBH is more efficiently measured than tree height, yet is strongly correlated with tree height.Therefore, the height-diameter models commonly applied in forestry are essential for the estimation of tree heights.Additionally, the estimation of biomass, stem basic density, growth and yield, and stand development over time in multilayered stands relies heavily on accurate height-diameter models (Huang et al. 1992;Temesgen et al. 2014;Nord-Larsen & Nielsen 2015;Sillett et al. 2019).It has also been shown that height-diameter models can even be used to estimate quantities of leaf, bark, cambium, sapwood and heartwood (Sillett et al. 2019).
For over 200 years, DBH has been widely used in forestry to estimate timber volumes, quantify ecosystem services, and to predict other biometrics that would be difficult to measure directly (Magarik et al. 2020).The most common practice for obtaining DBH is to measure trunk diameter at c. 1.3 m (4.3 ft) above ground level or at the interface between the trunk and root system.On sloping ground, this measurement is usually taken starting at the highest point of the soil touching the trunk.However, the actual height at which DBH is measured varies from region to region.For example, a DBH of 137 cm is accepted in the United States (corresponding to 4.5 feet in the imperial measurement system), while 130 cm is recognized in Europe and in tropical regions (Chave et al. 2014).
For living trees, different models have been proposed to determine the height-diameter relationships in different species, regions and ecological zones (Feldpausch et al. 2011;Temesgen et al. 2014;Chai et al. 2018), owing to multiple factors influencing tree growth.However, proposing specific height-diameter models is often not possible for ancient trees, because the fossil stems of woody plants are usually fragmentary, which makes the direct measurement of the heights of extinct plants only rarely possible (Thomas & Watson 1976;Stewart & Rothwell 1993;Taylor et al. 2009).The practical approach of modelling height-diameter relationship for ancient trees is based on the nearest living relatives of the fossil taxa.Up to now, three height-diameter models have commonly been applied to the height reconstruction of fossil trees.In 1990, Mosbrugger and colleagues proposed a power equation with an exponent of 0.67 for estimating the height of ancient trees using the material strength of wood based on Young's modulus and the specific weight of the wood of living Taxodium or Sequoia (Mosbrugger 1990;Mosbrugger et al. 1994).In 1994, the second approach was proposed by Niklas based on the manner in which woody stems taper in girth along their length; this resulted in a curvilinear equation for estimating overall fossil plant height (Niklas 1994).The third approach was taken five years later by Pole, who looked at the diameter and height data for three extant species of Araucaria growing in natural forests in Australia and New Guinea (Pole 1999) and used a power function with an exponent fixed to 0.67 for the height-diameter relationship.As the normalization constant of Pole's equation ( 1999) is about two to three times that of the Mosbrugger equation for conifers, Pole concluded that the heights of living trees are distinctly taller for any given diameter than the Mosbrugger curve would predict.All three approaches for the estimation of ancient tree heights were developed on the basis of tree architecture in living forests.
Here we gather height-diameter data of Araucaria from the scientific literature that originate from four living populations: A. bidwillii and A. cunninghamii in Queensland, Australia, and A. cunninghamii and A. hunsteinii in New Guinea.We then establish appropriate height-diameter growth models for each population and for the entire dataset by fitting and ranking 19 different nonlinear height-diameter models based on several model performance criteria.The best three growth models established for the genus Araucaria are each derived from 100 random samples of the entire dataset, and they pass multiple performance criteria.We then apply these three models to a group of fossil araucariaceous logs cropping out on the same stratigraphic horizon and compare predicted heights to the previously calculated heights of these Late Jurassic trees.

Datasets on Araucaria trees
We compiled datasets on tree height-diameter information from three Araucaria species: A. bidwillii (Pole 1999), A. cunninghamii (Gray 1975;Pole 1999) and A. hunsteinii (Paijmans 1970;Gray 1975).These datasets had originally been collected to determine the size composition of natural stands, describe forest structure, or analyse height-diameter relationships.However, standard in each of these studies was taking the diameter at breast height (DBH) of each tree trunk and the height (H ) of the respective tree.In our study, we used both measurements to establish the height-diameter growth models presented here.
In total, we collected and analysed the height-diameter data of 157 Araucaria trees.These trees grew in different forests of New Guinea and eastern Australia (Queensland).Data came from trees inhabiting a diversity of forest habitats with variable floristic composition and site conditions and exhibiting varying heights.To group Araucaria trees growing in the same region under roughly the same environmental conditions, each tree from our entire dataset of 157 trees was assigned to one of four populations (Table 1): A. bidwillii and A. cunninghamii in Queensland, and A. cunninghamii and A. hunsteinii in New Guinea.In Queensland, 54 trees of A. bidwillii and 47 trees of A. cunninghamii were measured in the Bunya Mountains National Park (Pole 1999).In New Guinea, A. cunninghamii (31 trees) and A. hunsteinii (25 trees) were analysed; they all inhabit primaeval forests (Paijmans 1970;Gray 1975).As a result, sample size differed substantially between populations.To preclude a bias of the height-diameter growth model(s) toward populations with the largest sample sizes, we modelled height against diameter not only for the entire dataset with 157 trees, but also for 100 random samples taken from this dataset.Each random sample consisted of all 25 trees of A. hunsteinii from New Guinea, 25 randomly selected trees of A. cunninghamii from New Guinea, 25 randomly selected trees of A. bidwillii from Queensland, and 25 randomly selected trees of A. cunninghamii from Queensland.

The testing of height-diameter growth models
To find the best height-diameter growth model for each of the four populations, for all 157 trees as well as for each of the 100 random tree samples, 19 height-diameter growth models were considered: a so-called modified Mosbrugger model which we introduce here, as well as the 2pPower, Niklas, von Bertalanffy, Näslund, Curtis, Schumacher, Meyer, Michaelis-Menten, Wykoff, Prodan, Logistic, Chapman-Richards, Weibull, Gompertz, Sibbesen, Korf, Ratkowsky and Hossfeld IV models (Table 2).All models are nonlinear and relate the height of an individual H (in metres) to its diameter D (in metres).They have been successfully applied to extant species, but their performance has rarely been statistically tested against each other for Araucaria species.Of the 19 models, 16 have already been used in other studies on height-diameter relationships in extant species (Table 2) (Huang et al. 1992;Mehtätalo et al. 2015;Chai et al. 2018), whereas the other three, namely, the modified Mosbrugger, Niklas and von Bertalanffy models, have not previously been statistically tested against these other models for trees.These three growth models are thus briefly introduced here.
Modified Mosbrugger height-diameter growth model.The original Mosbrugger height-diameter model is based on a physical core principle, that is, that any tree can be regarded as an upright and free-standing column that undergoes global buckling when loaded at the top (Mosbrugger 1990).Two parameters, Young's modulus and the specific weight of the column, are incorporated in the following equation (Eqn 1).
where L g , is the critical length (the free-standing column deforms by global buckling when length exceeds L g , in metres), C equals 0.85 if a tree is regarded as an upright and free-standing ideal column, E is Young's modulus of the column, w is the specific weight of the column, and r is the radius of the column (in metres).For a typical conifer, Mosbrugger et al. (1994) suggested that C and E/w equal 0.32 and 1.7 × 10 6 m, respectively.In this case, the E/w built into this equation was that of a conifer such as Taxodium or Sequoia (Mosbrugger et al. 1994).
The original Mosbrugger equation is in essence a 1parameter power function that relates stem length to stem radius, where the exponent is a constant equal to 0.67.Its normalization constant is given by C (E/w) 1/3 and is T A B L E 2 .Nonlinear height-diameter growth models selected for establishing growth models for the genus Araucaria.

Model name
Function type Equation References taxon-dependent.Thus, a revised formulation of the Mosbrugger model, called here the modified Mosbrugger model which we use in our study and fit to height-diameter records of Araucaria, is a power function that is parametrized by its normalization constant, whereas its exponent is fixed to 0.67 (Table 2).It not only simulates Mosbrugger's original formula (Eqn 1), but also that of Pole (1999) which is also a power function with an exponent of 0.67 and a normalization constant that was inferred from the three Araucaria species studied herein (Table 1).
Niklas height-diameter growth model.The Niklas heightdiameter model is a curvilinear function relating stem length measured from the tip of a stem to the stem diameter.This 3-parameter allometric model was used to describe growth in non-woody and woody species (Niklas 1994).For log-transformed stem lengths and diameters, the formulation of the Niklas model reads as follows: where L is stem length, d is stem diameter, β is the normalization constant, and both α 1 and a 2 are (allometric) exponents.For its polynomial formulation relating tree height H to DBH, see Eqn 2 and Table 2.
von Bertalanffy height-diameter growth model.The 2parameter von Bertalanffy growth model has been commonly used to describe growth in body size in relation to an individual's age.This asymptotic model has been applied to many animal taxa, including birds, snakes, lizards, turtles, crocodiles and extinct sauropod dinosaurs (Ricklefs 1968;Frazer & Ehrhart 1985;Halliday & Verrell 1988;Shine & Charnov 1992;Magnusson & Sanaiotti 1995;Lehman & Woodward 2008;Griebeler et al. 2013).The specific formulation of the von Bertalanffy model fitted to height-diameter records of Araucaria is shown in Table 2.
Establishment of growth models for the four populations and for the entire dataset All 19 height-diameter growth models (Table 2) were applied to each of the four Araucaria populations and to the dataset with all 157 trees in order to identify the best model(s) to predict height H from DBH for each.We did not account for differences in sample size of population in this analysis.For the Niklas model, its double logtransformed equation (Eqn 2) with log-transformed DBH and H values was used (Niklas 1994).For each Araucaria dataset, the best fitting model was identified from the Akaike's information criterion (using log-likelihood-based AIC values) and the AIC based model selection approach suggested by Burnham & Anderson (2002) (Burnham & Anderson 2002) for each of the four populations and for the entire dataset.

Establishment of growth models for the genus Araucaria
Only 10 of our 19 growth models passed our performance criteria for at least one of the four populations or for the entire dataset (see Results).To identify the best sample-size corrected growth model(s) for the genus Araucaria, each one of these 10 height-diameter models (Table 2) was fitted to each of the 100 randomly generated samples as described above.Based on the 100 curve fitting results obtained for each growth model and respective AIC values, we looked for the following two criteria to find statistical well-constrained models for the genus Araucaria.The first criterion was that for all 100 random samples, the fitting algorithm had converged for the model and that all parameter estimates obtained for the model differed significantly from zero (p ≤ 0.05).The second criterion was that parameter estimates and AIC values of the model showed little variation throughout the 100 random samples, which indicates that parameter estimates and goodness-of-fit are robust against the random sample used for curve fitting.Then, the so-called 'median models' were generated on the basis of the median values of their parameter estimates throughout the 100 random samples for each model meeting both criteria.
The three median models that satisfied our two criteria were then moved on to our final detailed evaluation.At this stage, their predictive performance was checked for each of the four populations and for the entire dataset with 157 trees.Specifically, we assessed how well each of the three median models predicted tree height H from DBH when compared to the model that was established for the respective dataset (e.g. a comparison between the modified Mosbrugger model obtained for A. bidwillii in Queensland and the median modified Mosbrugger model; the 2pPower model attained for the entire dataset of 157 trees and the median 2pPower model).Five criteria were used: the residual sum of squares (RSS; i.e. the total sum of squared residuals); the residual standard deviation (RSD; i.e. the standard deviation of the residuals); the residual standard error (RSE; i.e. the standard error of the residuals); the root mean square error (RMSE; i.e.RSS divided by the difference of sample size and the number of parameters used by the growth model); and the AIC value (RMSE-based).RSD and RSE evaluate the residual variation, while RSS, RMSE and AIC show overall model performance.The RSS, RSD, RSE, RMSE and AIC values were calculated for each of the median models and the respective growth model established for each height-diameter dataset.In the end, the overall best median model was the model that had the lowest RSS, RSD, RSE, RMSE and AIC values for a given height-diameter dataset.Consequently, the median models that best passed this final detailed evaluation were the height-diameter growth model(s) for the genus Araucaria selected and presented in our study.

RESULTS
Height-diameter growth models for the four populations and for the entire dataset We were able to apply successfully 13 out of 19 growth models (Table 2; i.e. the fitting algorithm revealed parameter estimates) to the four populations (using the maximum sample size on each one) and to the entirety of 157 trees growing in Queensland, eastern Australia, and in New Guinea (Tables S1-S5).Specifically, the modified Mosbrugger, 2pPower, Niklas, von Bertalanffy, Näslund, Curtis, Schumacher, Wykoff, Prodan, Chapman-Richards, Weibull, Sibbesen and Ratkowsky growth models worked for the four populations and their composite dataset, whereas the Meyer, Michaelis-Menten, Logistic, Gompertz, Korf and Hossfeld IV growth models were never applicable (i.e. the fitting algorithm did not converge and thus revealed no parameter estimates).However, for six models, at least one of the estimates on their parameters did not differ significantly from zero (p > 0.05; Tables S1-S5).For all populations and for the entire data of 157 trees, one model parameter estimate was always non-significant for the Ratkowsky, Chapman-Richards and Weibull growth models.Only for the A. cunninghamii population in Queensland was one parameter estimate of the Niklas model not significant.Parameter estimates of the Prodan model were always significant and identical across the four populations.The Schumacher growth model had also identical and significant parameter estimates for the population of A. hunsteinii in New Guinea and the entire population of 157 trees.Parameter estimates of the Prodan and Schumacher models were also significant and identical for the other three populations, but the parameter estimates came out different.Parameters estimated for all remaining growth models, that is, the modified Mosbrugger, 2pPower, Niklas, von Bertalanffy, Näslund, Curtis, Wykoff and Sibbesen growth models were significant and differed across the five datasets analysed.Overall, 10 growth models yielded significant estimates for all their parameters for at least one of the four populations or for the entire dataset.These 10 growth models (hereafter 'significant models') were the modified Mosbrugger, 2pPower, Niklas, von Bertalanffy, Näslund, Curtis, Schumacher, Wykoff, Prodan and Sibbesen models.
Araucaria hunsteinii population in New Guinea.Of these 10 significant models (Fig. 1; Table S1), the Näslund and Prodan models showed residual standard errors that were about a magnitude larger than those of the other models (Table S1) and did not generate any monotonic increase in tree height H with increasing DBH (Fig. 1).Of the other eight significant models, the modified Mosbrugger model produced the lowest AIC value (8.684).ΔAIC values less than 2 suggest that the von Bertalanffy (ΔAIC = 1.217),Curtis (ΔAIC = 1.288) and 2pPower (ΔAIC = 1.883) models are as well-supported as the modified Mosbrugger model.The ΔAIC values of the Wykoff (ΔAIC = 2.467), Niklas (ΔAIC = 3.197) and Sibbesen (ΔAIC = 3.217) models were also small, whereas that of the Schumacher model was much larger (ΔAIC = 6.970).
Comparatively high support was also found for the Wykoff (ΔAIC = 2.523) and Niklas (ΔAIC = 3.172) models, whereas the ΔAIC value (7.414) of the Sibbesen model was much larger.
Araucaria cunninghamii population in Queensland.For this population, only 9 of the 10 significant models were applicable (i.e. in the Niklas model, one parameter estimate was not significant; Fig. 3; Table S3).Of these, only the Näslund and Prodan models produced a residual standard error substantially larger than the other seven models (Table S3), and neither model yielded a monotonic increase in tree height H with increasing DBH (Fig. 3).Once again, the modified Mosbrugger growth model had the lowest AIC value (8.323) of the other seven models, yielding a monotonic increase in tree height.The ΔAIC values of the 2pPower (1.928) and von Bertalanffy (1.895) models were less than 2 and suggest that both models were as well-supported as the modified Mosbrugger model.Only the Wykoff model had a ΔAIC value close to 2 (2.265), whereas that of the Curtis (4.667), Sibbesen (4.844) and Schuhmacher (5.782) model were clearly larger (Table S3).
Araucaria bidwillii population in Queensland.Of the 10 significant models, only the Prodan model showed a residual standard error substantially larger than the other nine models (Fig. 4; Table S4), and the shape of its curve indicated no monotonic increase in tree height H with increasing DBH (Fig. 4).Once again, the modified Mosbrugger growth model attained the lowest AIC value (8.248).Four growth models had ΔAIC values less than 2, namely, the Näslund  Composite modelling of all Araucaria populations.Of the 10 significant models applied to the entire dataset of 157 trees, only the Prodan and Sibbesen models produced a residual standard error substantially larger than those of the other eight models (Fig. 5; Table S5), and the shape of their curves indicated a non-monotonic increase in tree height H with increasing DBH (Fig. 5).The modified Mosbrugger growth model had the lowest AIC value (10.753).Three growth models yielded ΔAIC less than 2 (the 2pPower (1.947), Curtis (1.948) and von Bertalanffy (1.965) models) suggesting that they were as wellsupported as the modified Mosbrugger model.The ΔAIC of the Näslund (2.120) and Wykoff models (2.172) were close to 2, whereas those of the Niklas (3.955) and Schumacher (4.935) models were clearly larger.
When comparing the modelling results of the four populations to that of all 157 trees, the modified Mosbrugger model turned out to be the best one for all five datasets.The 2pPower and von Bertalanffy models always produced a ΔAIC less than 2, while the Curtis model always yielded a ΔAIC smaller than 2, except in the case of A. cunninghamii in Queensland (4.667).The Wykoff model always had ΔAIC values around 2. The Schumacher model consistently showed a ΔAIC clearly greater than 2, except for the case of A. cunninghamii in New Guinea (1.469), while the Niklas model produced ΔAICs considerably greater than 2 for the four datasets for which its parameter estimates were significant (except for A. cunninghamii in Queensland).The Näslund, Sibbesen and Prodan models were the worst models for each of the four populations and for the entire dataset.The Näslund model had a residual standard error substantially greater than those of the other significant models and did not render a monotonic increase in tree height H with increasing DBH for two datasets (A.cunninghamii in Queensland and A. hunsteinii in New Guinea), while the Sibbesen model yielded a residual standard error substantially larger than that of the other significant models only F I G . 2 .Fitted significant height-diameter growth curves of Araucaria cunninghamii population in New Guinea (NG).
for the entire dataset, but its shape did not show a monotonic increase for the populations A. bidwillii in Queensland and A. cunninghamii in Queensland, nor for the entire dataset.The Prodan model always had a residual standard error substantially greater than those of the other significant models, and the shape of its curve did not always render a monotonic increase in tree height H with increasing DBH.

Height-diameter growth models for the genus Araucaria
Of the 10 significant models identified for the four populations and for the entire dataset of 157 trees, only 7 (modified Mosbrugger, 2pPower, von Bertalanffy, Curtis, Schumacher, Wykoff and Sibbesen models) yielded significant parameters for all 100 random samples (Table S6).However, the variability in parameter estimates and AIC values of the 100 random samples differed between these models (Fig. S1; Table S6).Curves derived from the 100 samples were the most variable for the Schumacher and Sibbesen models, and the Sibbesen model did not render a monotonic increase in tree height H with increasing DBH (Fig. S1).Although parameter variability was small for the Wykoff model, all 100 models substantially overestimated tree heights for small DBH (Fig. S1).The 100 modified Mosbrugger models showed the lowest variability in predicted heights across the DBH values exhibited by the 157 trees, which were followed by the 2pPower, von Bertalanffy and Curtis models (Fig. S1).Based on the median AIC values, these four growth models were similarly well-supported, although the modified Mosbrugger model attained the lowest (median) AIC (Table S6).The curves of the 100 von Bertalanffy models were very similar (Fig. S1), and the median AIC value across the 100 von Bertalanffy models was small (Table S6), however, we conducted our final detailed evaluation with only the modified Mosbrugger, 2pPower and Curtis median models.We justify this selection here because the estimated asymptotic heights (parameter a) of the von Bertalanffy model differed considerably across the four Araucaria populations (Tables S1-S4), which was reflected in the large variation that we observed in the 100 random samples for the parameter estimating asymptotic height (Table S6).
Our final, detailed evaluation of the three median models showed that the best fits in terms of RSS, RSD, RSE, RMSE and (RMSE-based) AIC were performed by the model respectively established for each of the four populations (Fig. 6; Table S7).There were only two exceptions: in the A. cunninghammii population in New Guinea, the RSD and RSE values of the modified Mosbrugger model were somewhat smaller than those of the median modified Mosbrugger model.For the full dataset on Araucaria, the differences in RSS, RSD, RSE, RMSE and AIC observed between median models and the respective ones established for the entire dataset were much smaller than that observed in the four populations (Table S7).
Contrary to our findings on the four populations and the entire dataset in which the modified Mosbrugger model always performed best, the RSS, RSD, RSE, RMSE and (RMSE-based) AIC values were smallest for the median 2pPower model and the 2pPower model in the entire dataset.Consistent with our results on the four populations (except for the A. cunninghamii population in Queensland, ΔAIC = 4.667 of the Curtis model) and the entire dataset (log-likelihood-based AIC values), the differences in (RMSE-based) AIC values smaller than 2 indicate that the median modified Mosbrugger, 2pPower and Curtis models were similarly well-supported.The values of the exponent of the 100 2pPower models ranged between 0.69 and 0.74, with a median value of 0.72.This indicates that the modified Mosbrugger model (exponent = 0.67) and the 2pPower model differ for the genus Araucaria.
F I G . 4 .Fitted significant height-diameter growth curves of Araucaria bidwillii population in Queensland (Qld), Australia.

DISCUSSION
Growth model performance Most of the selected height-diameter growth models were successfully fitted to the four populations, to the entire dataset (157 trees) and to 100 random samples of Araucaria.For the four populations and for the entire dataset of Araucaria, the modified Mosbrugger model turned out to be the best model for describing the height-diameter relationship of Araucaria trees on the basis of AIC-based model selection.However, the 2pPower and von Bertalanffy models were as well-supported as the modified Mosbrugger model based on the likelihood-based AIC values (ΔAIC < 2).Regarding the 100 random samples analysed to correct for differences in sample sizes between populations, the modified Mosbrugger, 2pPower and Curtis models were found to show the lowest variability in parameter estimates.They described the height-diameter relationships of Araucaria trees best and generated three median models: the median modified Mosbrugger, median 2pPower and median Curtis models.Based on multiple model performance criteria (RSS, RSD, RSE, RMSE, and (RMSE-based) AIC) the median 2pPower model was identified as the best median model in the final detailed evaluation of the three median models.Nevertheless, the median modified Mosbrugger and median Curtis models were nearly as well-supported as the median 2pPower model.
However, our results on the best growth models for the four populations, for the entire dataset and for the 100 random samples of Araucaria are inconsistent with previous studies on modelling height-diameter relationships of trees and the ranking of several of the 19 different height-diameter growth models mentioned above (Table 2; Huang et al. 1992;Mehtätalo et al. 2015;Chai et al. 2018).For example, Huang and colleagues found that Chapman-Richards and Weibull models best rendered the height-F I G . 5 .Fitted significant height-diameter growth curves of entire dataset of Araucaria.
F I G .6 .The curves of three median models on the genus Araucaria and curves derived from fitting the respective model to DBHheight records of the four populations and of the entire dataset of 157 trees.
diameter relationship of major Alberta species (angiosperms and conifers) in Canada (Huang et al. 1992).Mehtätalo et al. (2015) realized that the Näslund model provided a satisfactory fit in most height-diameter datasets of trees from Europe, Asia, North America and South America.Another study proposed that the Chapman-Richards, Weibull and Näslund models best performed in modelling height-diameter relationship of Cryptomeria fortunei trees (Cupressaceae) in the Guizhou Province of southwestern China (Chai et al. 2018).None of all these models worked sufficiently for any of our populations studied here (Tables S1-S5).An important reason behind this incongruence may be that the tree species modelled and large-scale and local-scale differences in growing conditions (e.g.temperature, precipitation, soil) influence height-diameter relationships.For example, the mature trees of Thuja plicata (western red cedar, Cupressaceae), a common species in the northwestern USA, obtain an average height of 40 m (25-52 m) when DBH equals 1 m (Van-derSchaaf 2013).On the other hand, in northern California, USA, the native coastal redwood Sequoia sempervirens (Cupressaceae) generally obtains tree heights of about 33-70 m, averaging about 50 m, when DBH equals 1 m (Sillett et al. 2019;Earle 2020;Vaden 2020).These two tree species document the large variability in tree growth due to general differences in height-diameter relationships between species and in local growing conditions within populations of species.They also show that cupressaceaous trees with the same DBHs would reach different heights due to variable growing conditions in differing regions and ecological zones.Further support for growing conditions affecting the height-diameter relationship at a large and local geographic scale comes from our study.Parameter estimates of the median 2pPower, modified Mosbrugger and Curtis models differ between A. cunninghamii from Queensland and New Guinea (Table S7), and there is a large variation in heights for any given DBH in all four populations (Figs 1-4).However, the strong species-effect on the best model (Table 2) in the extant taxa discussed above suggests that similar modelling studies on other extant tree genera sampled on a broad geographic scale are necessary in order to accurately reconstruct ancient tree height from growth models based on their closest living relatives.Such studies will also allow us to assess whether the effect of biological affinity on the species or genus-level generally remains strong.
We are aware that all these uncertainties involved in establishing our growth models for the genus Araucaria are greater than for each population of Araucaria.Nevertheless, we believe our approach for establishing most appropriate median models for the genus Araucaria is justified.Because the three median models obtained a small variation in parameter estimates and AIC values throughout the 100 random samples, the differences in (RMSE-based) AIC values between the median models and the respective model of the 157 trees were small (2.76-3.29).Nevertheless, it should be noted that the best model established for individual datasets is always better than the median models, because each dataset of trees experiencing different growing conditions reflects variable height-diameter relationships.

Pitfalls based on the preservation of fossil logs
Collecting precise diameter data on fossil logs is essential for the accurate estimation of ancient tree height, although there is commonly some deformation in the vertical plane of fallen tree trunks in the geological record.Four decades ago, Rex & Chaloner (1983) showed experimentally that cylindrical objects (e.g.fossil logs) are likely to be preserved in their original horizontal dimensions even when compressed vertically during geological burial.Another study based on the thin sections of fossil logs showed that compression is mostly accommodated by tracheids crushing or squashing in the parallel plane of compression (Falcon-Lang & Bashforth 2005).Hence, the width of a fossil log is assumed to approximate the original diameter of the ancient tree even though the fossil log may have been compressed vertically during geological burial.
In general, measuring trunk diameter where there is apparent damage or swellings near exposed branches, or near the root flare should be avoided (e.g.Williams et al. 2003).However, fossil logs found in fluvial sediments commonly lack their root system, as well as any evidence of root flare at the tree trunk-substrate interface, making the measurement of the exact DBH of fossil trunks impossible.Given that a trunk is not an ideal column but that diameter decreases along its height, using a diameter from a stump above breast height in a DBHheight model would underestimate tree height, while a diameter below breast height would overestimate tree height.In the end, the amount of underestimation and overestimation of tree height depends on the shape of the tree's trunk and how strongly the trunk deviates from an ideal column.
Bark is rarely completely preserved with fossil logs or trunks in deep time, because bark is easily detached from the central cylinder of secondary xylem after the tree has died or fallen (Staccioli et al. 1998).Bark thickness is strongly positively correlated with DBH (Pinard & Huffman 1997;Paine et al. 2010;Zeibig-Kichas et al. 2016).Although the ratio of bark thickness to DBH varies widely among families, an average ratio of 5% may generally be applied when reconstructing ancient tree heights (Pinard & Huffman 1997;Zeibig-Kichas et al. 2016).Neglecting bark layers generally leads to a slight underestimation of tree height based on DBH.

Case study revising the heights of Upper Jurassic araucariaceous trees
Fossil logs and woods are abundant in the Upper Jurassic Morrison Formation, a widespread terrestrial formation in the Western Interior of North America (Tidwell 1990;Ash & Tidwell 1998;Tidwell et al. 1998;Gee & Tidwell 2010;Gee et al. 2019Gee et al. , 2022Gee et al. , 2023;;Richmond et al. 2019;Sprinkel et al. 2019;Xie et al. 2021).Recently, Gee et al. (2019) described a local flora of fossil log and wood in this formation in Rainbow Draw in northeastern Utah, which represents a monospecific flora of Agathoxylon hoodii.The fossil logs measured from 71 to 127 cm in preserved diameter (Gee et al. 2019).Using the original formula proposed by Mosbrugger (1990) and Mosbrugger et al. (1994) to estimate ancient tree height (Eqn 1), Gee et al. (2019) reconstructed the Rainbow Draw trees based on their maximum preserved diameters of 0.71-1.27m as reaching minimum heights of 19-28 m (Table 3).
To recalculate the heights of the araucariaceous trees in Rainbow Draw using the same diameter data of Gee et al. (2019), we apply here the three best models for the genus Araucaria, the median modified Mosbrugger (Eqn 3), median 2pPower (Eqn 4) and median Curtis (Eqn 5) models.
The modified Mosbrugger median model gives new reconstructed height values of the Rainbow Draw trees from 46 to 67 m, while the 2pPower median model reconstructs the tree heights as ranging from 45 to 69 m, and the Curtis median model estimates the tree heights as 47-67 m (Table 3).Our new results for reconstructed heights of Rainbow Draw trees are very consistent across all three models (Table 3).
Thus, using the three median models, the original estimates by Gee et al. (2019) of heights between 19 and 28 m for the Rainbow Draw logs are revised here to heights between c. 45 and 70 m, which is substantially greater than those originally predicted.The application of the original formula of Mosbrugger et al. (1990Mosbrugger et al. ( , 1994) ) underestimated tree height of the Rainbow Draw logs by about two and a half times.Consistent with our new estimates for the Rainbow Draw logs, Pole (1999) observed that the heights of living Araucaria trees are distinctly taller (c.2-3 times) for any given diameter than the Mosbrugger curve would predict.His observation and our modelling study most probably suggest that the safety factor of 0.5 built into the original formula of Mosbrugger et al. (1990Mosbrugger et al. ( , 1994) ) for conifers such as Taxodium or Sequoia is unnecessary, at least for Araucaria species (see Mosbrugger 1990).When comparing the original Mosbrugger formula to that of our median modified Mosbrugger model (Eqn 3), the normalization constant of our median model corresponds to C (E/w) 1/3 (Eqn 2).Thus, we cannot rule out that the safety factor is correct for Araucaria.Our higher estimates of tree heights could also indicate larger E/w values (due to a difference in both constants, a larger Young's modulus E (no difference in w), or a smaller specific weight of the column w (no difference in E )) in Araucaria species compared with those of Taxodium or Sequoia.

CONCLUSION
In this study, 19 nonlinear growth models were selected to fit height-diameter relationships in four populations, the entire dataset, and 100 random samples of Araucaria growing in New Guinea and Queensland.For each of the four populations and for the entire dataset of Araucaria, the modified Mosbrugger model was the best model to describe height-diameter relationships in trees, but the 2pPower and von Bertalanffy models were as wellsupported as the modified Mosbrugger model in terms of AIC values.From the analysis of 100 random samples, three growth models with small parameter-estimate variations and small AIC values were identified, which then generated three median models.Specifically, the median 2pPower model was the best median model to describe the height-diameter relationship of the genus Araucaria; the median modified Mosbrugger and median Curtis models performed nearly as well as the median 2pPower model.A case study in which the heights of araucariaceous logs in the Upper Jurassic Morrison Formation in Rainbow Draw, Utah, were recalculated shows that tree height increases by about two and a half times when these median models were applied (i.e. from a maximum height of 28 m to 70 m).This difference in heights of araucariaceous logs was most likely introduced by the safety factor (0.5) built into the original Mosbrugger formula.

F
I G . 1 .Fitted significant height-diameter growth curves of Araucaria hunsteinii population in New Guinea (NG).
. Specifically, out of all 19 height-diameter growth models fitted, only those models with significant parameter estimates (i.e. each parameter estimate of a model differs significantly from zero) found for the dataset were ranked based on their AIC values, and the model with the lowest AIC (min(AIC)) was identified.ΔAIC (AIC À min(AIC)) was then calculated for each model.ΔAIC scores less than 2 suggest similar well-supported models; ΔAIC scores ranging from 2 to 10 indicate a moderate support of the model with the lowest AIC over these models; ΔAIC scores greater than 10 indicate that the respective model is weakly supported compared to the model with the lowest AIC (with the lowest AIC).Differences in model performance were further assessed by AIC-based Akaike weights Gee et al. (2019)sed heights of Upper Jurassic araucariaceous trees from Rainbow Draw in Utah, USA.Original specimen designations and diameters fromGee et al. (2019).Eqns 3-5 were used to calculate the values resulting from median models.RD, Rainbow Draw; FHNHM, Utah Field House of Natural History State Park Museum in Vernal, Utah.