Growth Models for Even-Aged Stands of Hesperocyparis macrocarpa and Hesperocyparis lusitanica

Abstract

The cypress species Hesperocyparis lusitanica (Mill.) Bartel and H. macrocarpa (Hartw.) Bartel (formerly known as Cupressus lusitanica and C. macrocarpa) are widely grown in New Zealand where they are noted for their high-value timber. Preliminary cypress growth models were developed in New Zealand in 2004 based on limited growth data. Here, we describe new stand-level growth models developed for these two species using measurements from 521 permanent sample plots. Each model consists of several component sub-models including a height/age model, a diameter/age model, a mortality function and a stand-level volume function, with different parameter estimates used for each species. The mean top height is predicted from age and site index using common-asymptote Richards models. Modified common-asymptote Korf functions are used to predict the mean diameter as a function of age, stand density and a diameter index. The volume-based 300 Index, defined as the mean annual volume increment at age 30 years for a reference regime of 300 stems ha⁻¹, can be calculated directly from the site index and diameter index using the volume function. These models will be of considerable value to forest managers for regime evaluation and yield prediction. By more robustly describing site productivity using the 300 Index, the models provide an improved framework for understanding and quantifying site productivity.

1. Introduction

Understanding plantation growth and yield is of vital importance to forest managers and planners for optimising decisions around site selection and the management of forest resources [1,2]. Traditionally, these decisions have been supported by growth models, which can be broadly categorised as empirical, hybrid or process-based in structure [2]. Each of these model types has strengths and weaknesses. Although empirical models are the most widely used for operational planning, the last few decades have seen a proliferation of hybrid and process-based models, many of which are now emerging as useful tools for forest managers [3].

As empirical models are fitted statistically using historical datasets, they closely reflect past observations, and when fitted to extensive datasets, they can often very accurately simulate growth. However, as such models are not grounded in the processes that determine growth, they are of limited use when applied beyond the conditions of the base data, into, for instance, spatial or temporal domains with different climates [1,4]. At the other end of the continuum, process-based models predict productivity as a function of the most important physiological processes underpinning growth and mortality. Often, these models are developed from intensive studies that mathematically characterise the complex myriad of processes that influence growth and the impact of environment on these relationships [5,6,7]. Process-based models often require extensive parameterisation, but when rigorously tuned to a species, they can be used with greater confidence than empirical models to understand and explore system behaviour [1].

Through integrating components from these two model classes, hybrid models are simpler to parameterise than process-based models but have greater generality than empirical models. Although there are a wide range of model formulations within this category [8], most hybrid models integrate the key influences of the environment into a traditional empirical structure through the inclusion of a growth modifier or environmental covariates [9,10,11]. The performance of hybrid models has often been found to be superior to empirical models, particularly at the stand-level [12,13,14], but this varies with the model resolution, species and type of hybridisation [2]. Importantly, hybrid models provide a framework that can incorporate sensitivity to climate and environment into the model structure, which often allows predictions to be made under climate change [8] and in regions beyond the range of the base data [15].

A weakness of many empirical modelling approaches is the way in which site quality is integrated into the model. Site quality is typically incorporated through the site index, SI, defined as the mean height of dominant trees (e.g., mean top height, MTH) at a base age. The site index was originally selected in the late 19th century to represent site quality [16,17], later justified by Eichhorn’s rule [18], which hypothesised a strong relationship between volume and height in even-aged stands regardless of stand density or site. However, research over the last 70 years clearly shows that environment and stand density have a differential influence on height and basal area. As a consequence, there is often considerable variation in volume for stands of the same height [19,20,21,22,23,24,25,26,27,28], which can reach up to 30% [29]. Despite this advance in our understanding, SI is still the most widely used proxy for site quality in empirical models, which limits the sensitivity of volume predictions to environment within these models.

Site quality can be more robustly integrated into empirical models through the use of indices based on both height and volume. As volume is affected by stand density and thinning history as well as site, the index needs to be adjusted for these silvicultural factors and age so that it represents a genuine index of productivity. Previous research has developed growth models based on a volume index from permanent sample plot (PSP) data for Pinus radiata D. Don (radiata pine) [27] and Sequoia sempervirens (Lamb. Ex D.Don) Endl. (coast redwood) [28] growing in New Zealand. Within these two models, MTH is predicted as a function of stand age and SI, and a series of equations is used to describe the tree quadratic mean diameter (D_q) as a function of age, stand density (N) and thinning effects. The impacts of site quality on growth are integrated into the model of D_q through an index (D₃₀) that describes diameter productivity for a reference regime of 300 stems ha⁻¹ at age 30 years. Mortality is predicted as a function of age and D_q, and a stand-level volume function predicts the volume from the basal area and MTH.

Although the D_q model is formulated using D₃₀, a more useful volume-based index of site productivity is provided by the 300 Index, defined as the mean annual volume increment at age 30 for a reference regime of 300 stems ha⁻¹, which can be derived directly from SI and D₃₀ using a volume function. Using this approach, previous research has estimated the 300 Index and SI for radiata pine and redwood growing within New Zealand from extensive PSP datasets. These data have then been used to develop fine-resolution surfaces of the 300 Index and SI for both species using a multitude of environmental covariates describing climate, edaphic properties and landform [30,31]. By linking these productivity surfaces to the growth models, they are effectively transformed into hybrid models sensitive to fine-scale environmental variation.

These growth models, which predict D_q as a function of a diameter productivity index, D₃₀, age, N and thinning effects, can be contrasted with basal area (BA) projection models which predict BA from age and an initial BA at some starting age. In such BA models, the initial BA provides the local site information. This BA modelling methodology is identical to the usual approach for modelling height growth. However, although it is straightforward to reformulate such BA models to provide local site information using BA at a standard base age similar to the way SI is used in height models, this is of little value because BA is strongly affected by stand density and thinning history as well as site productivity. In addition, such models predict that future BA growth will be identical for a given initial BA and age, regardless of stand density, which may not necessarily be true. A BA projection model of this type has been developed in New Zealand for the two cypress species [32], which are the subject of this study, providing a benchmark against which the new modelling approach can be tested.

Cypresses are evergreen coniferous species belonging to the family Cupressaceae that are native to northern temperate regions. Within New Zealand, cypresses have been planted for over 150 years, although they constitute only 0.6% (9928 ha) of the 1.7 million ha plantation resource compared with the 90% occupied by radiata pine [33]. The most widely established cypress species in New Zealand are Hesperocyparis lusitanica (Mill.) Bartel (Mexican cedar, formerly named Cupressus lusitanica) and Hesperocyparis macrocarpa (Hartw.) Bartel (Monterey cypress, formerly named Cupressus macrocarpa). The natural range of H. lusitanica is quite extensive, covering central Mexico and Guatemala at elevations of 1200–3000 m [34]. Over its natural range, it grows in isolated pockets rather than continuous stands, usually on moist slopes or near streams. In contrast, H. macrocarpa probably has the most restricted natural range of any conifer in the world, being confined to two groves totaling only 30 hectares on a coastal site in Monterey, California [34]. These two species produce naturally scented, high value, appearance-grade timber that is naturally durable and dimensionally stable with an attractive grain. The timber can be used for exterior cladding and joinery, interior mouldings and paneling, boat building and furniture making [32,35]. Both species are affected in New Zealand by cypress canker disease caused by two fungi, Seiridium cupressi (Cooke and Ellis) Sutton and S. cardinale (Wagener) Sutton and Gibson, which affects branch tips, destroys stem cambium and can result in fluting and occasional tree death [36,37,38]. Hesperocyparis lusitanica has a markedly lower susceptibility to cypress canker than H. macrocarpa and consequently is often the preferred species in warm areas where canker is more severe [36,39,40].

Growth equations have been developed for H. lusitanica growing in central Amrica [41,42], eastern Africa [43,44,45,46] and southern Brazil [47]. In New Zealand, preliminary growth models based on limited data were derived for both H. lusitanica and H. macrocarpa in 2004 [32]. However, as considerable additional growth data is now available, it is timely to develop new growth models for both species using the 300 Index modelling approach. This should allow site productivity to be better quantified than is possible using SI, the only index of site productivity provided by the earlier models. The development of these growth models will allow variation in growth between the two cypress species to be evaluated across different sites. The objective of this study was to develop stand-level growth models for both cypress species using the 300 Index modelling methodology. This modelling approach incorporates functions to predict MTH, D_q, N, BA and stem volume across a range of thinning regimes and stand densities. The fit and structure of these models is described, and growth predictions are compared between the two cypress species.

2. Materials and Methods

2.1. Data

The data used to develop the cypress growth models consisted of 3207 measurements (1626 H. macrocarpa and 1581 H. lusitanica) from 521 PSPs (355 H. macrocarpa and 166 H. lusitanica). This dataset was much more extensive than that used by Berrill [32], who developed a preliminary stand-level growth model for cypress in New Zealand using the then-available data from 166 H. lusitanica and 163 H. macrocarpa PSPs.

The PSPs available for the current study were located at 334 sites (260 H. macrocarpa and 69 H. lusitanica, with 5 sites having both species represented). Plots were considered to be from the same site when they were in close proximity (mostly less than one kilometre) with planting year differing by at most one year. Some 281 sites contained only a single growth plot, and these provided representative growth data with a wide geographic dispersion. Twenty-seven other sites contained plots with varying stand densities, providing information on how growth varied with stand density at the same site. In most cases, these sites consisted of two to four stand densities without replication, but there were also several larger trials containing multiple replications. The remaining 26 sites contained multiple plots but with similar stand densities.

Plot area and measurement age were recorded along with diameter at 1.4 m breast height (DBH, cm) for all stems and stem heights for a sample of trees. Plot area averaged 0.043 ha (range 0.008–0.25 ha). Age ranged from 2–77 years, while stem volume averaged 229 and 178 m³ ha⁻¹ for H. macrocarpa and H. lusitanica, respectively, and MTH averaged 14.8 and 14.4 m for H. macrocarpa and H. lusitanica, respectively (

Table 1. Summary of permanent sample plot data used to develop cypress growth models.

Variable	Hesperocyparis macrocarpa			Hesperocyparis lusitanica
	Mean	Std. dev.	Range	Mean	Std. dev.	Range
Age (t, years)	18.1	12.9	2–77	13.8	7.7	2–61
Mean top height (MTH, m)	14.8	7.3	1.6–41.2	14.4	6.0	2.3–35.9
Basal area (BA, m2 ha−1)	32.3	24.0	0.0–178.1	27.5	16.9	0.0–117.3
Stem volume (V, m3 ha−1)	229	269	0–2435	178	165	0–1282
Quadratic mean breast height diameter (Dq, cm)	24.6	11.9	1.0–74.3	24.3	10.3	0.7–57.6
Stand density (N, stems ha−1)	764	485	32–4950	650	371	104–2386

). Plots were well distributed across New Zealand (Figure 1), covering a wide range in derived (see later methods) SI, defined as MTH (mean top height) at age 30 years (range 8.8–33.6 and 12.5–36.1 m for H. lusitanica and H. macrocarpa, respectively) and 300 Index (range 0.44–29.6 and 0.73–31.3 m³ ha⁻¹ year⁻¹ for H. lusitanica and H. macrocarpa, respectively). Similar growth rates have been recorded for H. lusitanica in southern Brazil, with mean annual volume increments ranging from 6–31 m³ ha⁻¹ year⁻¹ at ages of 16–18 years [47], while somewhat lower yields of 6–17 m³ ha⁻¹ year⁻¹ were reported in Ethiopia at ages 25 to 34 years [43].

2.2. Analysis Methods

The growth models developed in this study consist of several component sub-models including MTH/age models, which predict MTH as a function of stand age with site quality controlled by SI, DBH/age models, which predict D_q as a function of age and N, using an index of DBH productivity, D₃₀, to account for site, mortality models, which predict N as a function of age and D_q, and a stand-level volume function predicting V from MTH and BA. The volume-based 300 Index can be calculated directly from SI and D₃₀ using the volume function providing a more useful index of site productivity than either SI or D₃₀. These sub-models are described in the following subsections.

2.2.1. Calculation of Stand Metrics

The following stand metrics were calculated for each plot measurement: stand density (N, stems ha⁻¹), mean top height (MTH, m), basal area (BA, m² ha⁻¹), quadratic mean DBH (D_q, cm) and under-bark stem volume (V, m³ ha⁻¹). MTH was obtained by fitting a Petterson Type 1 height/DBH equation for each plot measurement and using this equation to predict height at the quadratic mean DBH of the 100 largest diameter stems per hectare [48]. Stem volume was calculated from measured or predicted height and DBH of each stem using an existing cypress volume function (function T280, developed using sectional measurements of 108 cypress trees in New Zealand, DBH range 7–84 cm, height range 6–38 m). All calculations were performed using R [49].

2.2.2. MTH/Age Model

Three sigmoidal growth functions were tested for predicting MTH from stand age (t, years), these being the Richards model [50] (also known as the Bertalanffy–Richards or Chapman–Richards model), the Hossfeld IV model [51] and the Korf model [52]. Both anamorphic and common-asymptote forms of each model were tested. In addition, for the Korf model, an advanced polymorphic form derived using the GADA method [53], with properties intermediate between the anamorphic and common-asymptote forms, was tested. Details of derivations of these models are provided in the Supplementary Material. Their equations are shown in

Table 2. Models tested for predicting stand-level variable, y, as a function of stand age, t. The base forms of each model have model parameters a, b and c. Anamorphic forms and the Korf GADA form have parameters b and c, while common-asymptote forms have parameters a and c. In both cases, models include initial values of y, y₀, at age t₀.

Base model	Form	Equation
Richards	Base form	$y=a{\left(1-exp\left(-bt\right)\right)}^c$
	Anamorphic	$y=y_0{\left(\left(1-exp\left(-bt\right)\right)/\left(1-exp\left(-b{t}_0\right)\right)\right)}^c$
	Common-asymptote	$y=a{\left(1-{\left(1-{\left(y_0/a\right)}^{1/c}\right)}^{t/t_0}\right)}^c$
Hossfeld IV	Base form	$y=t^c/\left(b+t^c/a\right)$
	Anamorphic	$y=t^c/\left(b+t^c\left(1/y_0-b/t_0^c\right)\right)$
	Common-asymptote	$y=t^c/\left(t_0^c/y_0+\left(t^c-t_0^c\right)/a\right)$
Korf	Base form	$y=aexp\left(-b{t}^{-c}\right)$
	Anamorphic	$y=y_{0}exp\left(-b{t}^{-c}\right)/exp\left(-b{t}_0^{-c}\right)$
	Common-asymptote	$y=a{\left(y_0/a\right)}^{{\left(t/t_0\right)}^{-c}}$
	Polymorphic GADA	$y=exp\left(R/2-2b/\left(R{t}^c\right)\right)$ where, $R=ln{y}_0+\sqrt{{\left(ln{y}_0\right)}^2+4b/t_0^c}$

.

To model MTH, an intercept of 0.3 m was included in all models representing the typical planting height of nursery-raised seedlings. The dependent variable was therefore y = MTH − 0.3. Local site effects were incorporated into all models using the parameter y₀ = SI − 0.3, where SI was defined as MTH at base age t₀ = 30 years. All models were fitted as nonlinear mixed models using the R nlme procedure, with y₀ fitted as a random effect with mean representing the mean of (SI − 0.3) across all plots and with a nested variance structure (plot within site). Thus, for example, the common-asymptote Richards model was fitted using:

$$MT{H}_{ijk}=0.3+{\left(1-{\left(1-{\left(\left({\mu }_{SI}-0.3+s_i+p_{ij}\right)/\left(a-0.3\right)\right)}^{1/c}\right)}^{t_{ijk}/30}\right)}^c+e_{ijk}$$

(1)

where MTH_ijk is mean top height of measurement k, plot i, site j, t_ijk is its age, µ_SI is the overall mean SI, s_i is a random effect representing the deviation of SI from µ_SI for site i (assumed to be normally, independent and identically distributed (niid)), p_ij is a random effect representing the deviation in SI from the site mean for plot j (assumed niid), e_ijk is a random error term (assumed niid) and a and c are fixed-effect model parameters. Model goodness of fit was evaluated using the Akaike Information Criterion (AIC). Models were fitted to combined data from both species and also separately for each species. Likelihood ratio tests were used to determine whether species-specific parameters provided a significantly better fit than the combined model. By inverting the equation of the best fitting model, SI was estimated for each plot from the final measurement, as was the age t_1.4 when the plot achieved a MTH of 1.4 m (breast height). Details of these procedures are given later in the paper.

2.2.3. DBH/Age Model

Key components of the cypress growth models are equations for predicting D_q from stand age. The general form of these models is similar to a recently developed redwood model [28] and an earlier radiata pine model [27]. To derive the D_q models, the first step was to test the model forms listed in Table 2 for predicting D_q as a function of breast height age, t_BH = t − t_1.4 (years), used in preference to age from planting to ensure that predicted D_q was zero at 1.4 m MTH. Local site effects were incorporated into these models using the parameter y₀ = D₃₀, with D₃₀ defined as D_q at base age t₀ = 30 − t_1.4 years (i.e., 30 years after planting). All models were fitted using the R nlme procedure with a nested variance structure (plot within site). The best-fitting models were chosen based on the AIC and used in subsequent steps.

The models fitted in Step 1 did not include N in their formulation, and the next step was to incorporate this. The function used to achieve this was similar to functions used in the redwood [28] and radiata pine [27] 300 Index models and had the following form:

$$D_q=D\left(t_{BH}\right)-\left(d/f\right)\left(lnN-ln300\right)ln\left(1+exp\left(f\left(D\left(t_{BH}\right)-\left(g_1+g_{2}lnN\right)\right)\right)\right)$$

(2)

where D(t_BH) is the predicted D_q for a stand growing at a density of 300 stems ha⁻¹, using the best fitting sigmoidal model from Step 1, N is stand density (stems ha⁻¹) and d, f, g₁ and g₂ are model parameters. The rationale underlying the form of this model is explained in [28]. Briefly, at young ages, the second part of the RHS of Equation (2) is close to zero, meaning that D_q closely follows the reference growth curve $D\left(t_{BH}\right)$. However, at older ages, the D_q growth curve trends below the reference growth curve for stand densities greater than 300 stems ha⁻¹, and above the reference curve for stand densities less than 300 stems ha⁻¹. This represents the effects of competition, with growth slowing at stand densities greater than the reference density. The midpoint of this transition occurs at D_q = g = g₁ + g₂lnN, and the rapidity of the transition is controlled by the parameter f. In cypress, it was found that competition began at smaller values of D_q at higher stand densities. This effect was accounted for by expressing g as a function of logN. Thus, for example, the common-asymptote Korf model was fitted using:

$$D_q{}_{ijk}=D\left(t_{BH}{}_{ijk}\right)-\left(d/f\right)\left(ln{N}_{ijk}-ln300\right)ln\left(1+exp\left(f\left(D\left(t_{BH}{}_{ijk}\right)-\left(g_1+g_{2}ln{N}_{ijk}\right)\right)\right)\right)+e_{ijk}$$

(3)

where $D\left(t_{BH}{}_{ijk}\right)=a{\left(\left({\mu }_{D_{30}}+s_i+p_{ij}\right)/a\right)}^{{\left(t_{BH}{}_{ijk}/\left(30-t_{1.4}{}_{ij}\right)\right)}^{-c}}$.

Here, $D_q{}_{ijk}$ is the quadratic mean DBH of measurement k, plot i, site j, $t_{BH}{}_{ijk}$is its breast height age, N_ijk is its stand density, $t_{1.4}{}_{ij}$. is the age the plot achieves 1.4 m MTH, ${\mu }_{D_{30}}$is the overall mean of D₃₀, s_i is a random effect indicating the deviation in D₃₀ from the overall mean for site i (assumed niid), p_ij is a random effect indicating the deviation in D₃₀ from the site mean of plot j (assumed niid), e_ijk is a random error term (assumed niid) and a, c, d, f, g₁ and g₂ are fixed-effect model parameters. A likelihood ratio test was used to test the statistical significance of the improvement in fit of this model compared with the best-fitting Step 1 model. Models were fitted to the combined data from both species and also separately for each species. Likelihood ratio tests were used to determine whether species-specific parameters provided a significantly better fit than the combined model.

2.2.4. Incorporating Thinning Effects into the DBH Model

The model described in Section 2.2.3, which predicts D_q as a function of the site productivity index, D₃₀, t_BH, and N, cannot be directly applied to thinned stands because the change in N at thinning would result in a discontinuity in the predicted D_q. To overcome this, the method adopted in the redwood and radiata pine models was used. When a stand is thinned from N₁ to N₂ stems ha⁻¹ at breast height age t_thin, there is often a selection effect due to smaller stems being thinned, and the ratio of the basal area after thinning, BA₂, to the basal area before thinning, BA₁, is usually greater than the ratio N₂/N₁. This effect is expressed by the coefficient p being less than 1 in the following equation:

$$B{A}_2/B{A}_1={\left(N_2/N_1\right)}^p$$

(4)

For an individual PSP thinning event, p can be estimated using:

$$p=\left(lnB{A}_2-lnB{A}_1\right)/\left(ln{N}_2-ln{N}_1\right)$$

(5)

The cypress PSP data contained records of 240 thinning events, enabling an average value of p to be estimated with good precision.

Using Equation (4), the following relationship can be derived for predicting D_q following thinning, D₂, from its pre-thin value, D₁:

$$D_2=D_1{\left(N_1/N_2\right)}^{\left(1-p\right)/2}$$

(6)

In a thinned stand, Equation (2) is used to predict D₁ prior to thinning using the pre-thin stand density, and D₂ is then calculated using Equation (6). An iterative procedure [28] is then used to determine the age, t’, when D_q is predicted to equal D₂ using Equation (2). The difference Δt_thin = t’ − t_thin is then calculated. If positive, Δt_thin represents the loss in development time because of competition at the higher pre-thin stand density, although in young stands Δt_thin can be negative due to the thinning selection effect. The growth in D_q is predicted following thinning by Equation (2) using t_BH − Δt_thin in place of t_BH. As in both the radiata pine and redwood models, this procedure is refined slightly by increasing Δt_thin by an additional small amount to account for crown recovery following thinning (based on the radiata pine model, Δt_thin is increased by 50% to a maximum of 0.25 years).

Many of the plots in the measurement data contained thinning events. Rather than attempting to adjust for these when fitting the model, it was first fitted to all data and Δt_thin was calculated for each measurement. All measurements where Δt_thin was greater than one year were then excluded from the data, and the model was refitted to produce the final D_q model.

2.2.5. Mortality Model

Mortality in forest plantations can include attritional mortality, catastrophic mortality caused by extreme events such as droughts or storms and competition-induced mortality. Constant attritional mortality can be predicted using the following model:

$$N_1=N_0{\left(1-k\right)}^{\Delta t}$$

(7)

where N₀ and N₁ are stand densities (stems ha⁻¹) at the beginning and end of a measurement increment, respectively, Δt is the increment length in years and k is a parameter representing the annual mortality rate. This model does not account for competition-induced mortality. A measure of competition can be obtained using Reineke’s stand density index [54], which, for metric measurements, is calculated using:

$$SDI=0.002288N{D}_q^{1.6}$$

(8)

where D_q (cm) is the quadratic mean DBH and N (stems ha⁻¹) is the stand density. It can be reasonably assumed that competition-induced mortality increases with SDI, suggesting a modification to Equation (7) as follows [28]:

$$N_1=N_0{\left(1-k-m{\left(SDI/1000\right)}^n\right)}^{\Delta t}$$

(9)

where k is a parameter representing attritional mortality and n and m control competition-induced mortality based on SDI (divided for convenience by 1000). Equations (7) and (9) were fitted to all measurement increments in the data using the R gnm procedure with the logistic link and quasi-binomial error function. Because this model is intended to be implemented using annual step lengths, the value of SDI used when fitting the model was determined at age $t+\Delta t/2-1/2$ using linear interpolation.

2.2.6. Stand-Level Volume Function

A stand-level volume function of a form used for radiata pine [55] and redwood [28] was fitted to predict the under-bark stem volume for each plot measurement from BA and MTH. As the standard deviation of the residual was proportional to the mean, the volume was log-transformed, and the function was fitted using the R nls function:

$$ln\left(V_i\right)=ln\left(MT{H}_{i}B{A}_i\left(v{\left(MT{H}_i-1.4\right)}^{-w}+u\right)\right)+e_i$$

(10)

where V_i (m³ ha⁻¹), BA_i (m² ha⁻¹) and MTH_i (m) are stem volume, BA and MTH, respectively, for plot/measurement i, while u, v and w are model parameters and e_i is a niid error term. The model was fitted using all measurements with MTH greater than 5 m.

2.2.7. Estimation of the 300 Index from a Plot Measurement

The 300 Index can be estimated from a plot measurement as follows. Firstly, SI is estimated from MTH (details presented later in the paper). D₃₀ is then estimated using an iterative procedure from the plot age, BA, SI and N together with thinning history. This procedure is described in the Supplementary Material of [28]. The volume function (Equation (10)) is then used to estimate the 300 Index, I₃₀₀, from SI and D₃₀ using the volume function, i.e.:

$$I_{300}=\left(300\pi SI{\left(D_{30}/200\right)}^2\right)\left(v{\left(SI-1.4\right)}^{-w}+u\right)/30$$

(11)

Note also that D₃₀ can be calculated from SI and I₃₀₀, allowing the growth models to be applied for any SI and 300 Index as follows:

$$D_{30}=200\sqrt{30I_{300}/\left(300\pi SI\left(v{\left(SI-1.4\right)}^{-w}+u\right)\right)}$$

(12)

2.2.8. Model for Predicting MTH from Mean Height

The following model for predicting MTH from mean height and stand density [28] was fitted using the R nls procedure:

$$MT{H}_i=H_{mean}{}_i/\left(1+r\left(1-exp\left(s\left(N_i-100\right)\right)\right)\right)+e_i$$

(13)

where MTH_i (m), $H_{mean}{}_i$ (m) and N_i (stems ha⁻¹) are MTH, mean height and stand density, respectively, for plot/measurement i, while r and s are model parameters and e_i is a niid error term. This model was used to predict MTH when it was not directly measured.

2.2.9. Model Testing

The growth models were tested using an Excel VBA implementation. Because of the limited New Zealand growth data for these species, all available data were used to develop the models, and validation against independent data was not possible. However, the performance of the models was tested using the development data, tabulating the predicted and actual means against a range of factors and plotting actual vs. predicted values and comparing these with predictions using the earlier Berrill growth model [32]. The models were also implemented in R code, which is included in the Supplementary Material.

3. Results

3.1. MTH/Age Model

Based on the AIC, the common-asymptote forms of the models for predicting MTH provided better fits than the anamorphic forms, while for the Korf model, the common-asymptote form also fitted better than the polymorphic GADA form (

Table 3. Akaike Information Criterion (AIC) of nonlinear mixed models for predicting MTH from stand age.

Model	Form	AIC
Richards	Anamorphic	8558.3
	Common-asymptote	8338.8
	Common-asymptote, separate parameters for each species	8094.9
Hossfeld IV	Anamorphic	10,504.9
	Common-asymptote	8314.0
	Common-asymptote, separate parameters for each species	8095.0
Korf	Anamorphic	8506.4
	Common-asymptote GADA	8344.9 8385.6
	Common-asymptote, separate parameters for each species	8138.6

). Of the three base model tested, the Hossfeld IV model was slightly superior to the Richards or Korf models when fitted to the combined species data, but when separate parameter estimates were fitted for each species, the Richards model was marginally superior and also fitted significantly better than the combined species model (likelihood ratio test, ${\chi }_6^2$= 225.9, p < 0.0001). Parameter estimates for the model fitted separately for each species (

Table 4. Parameter estimates of common-asymptote Richards MTH models (Equation (14)) fitted using the model described by Equation (1). See description of Equation (1) for parameter definitions.

Parameter		H. lusitanica				H. macrocarpa
		Estimate	Std. Error	t-Value	p-Value	Estimate	Std. Error	t-Value	p-Value
Fixed effects	µSI	24.52	0.56	43.8	<0.0001	22.19	0.25	58.8	<0.0001
	a	38.79	0.59	65.8	<0.0001	47.37	1.53	31.0	<0.0001
	c	1.099	0.132	8.33	<0.0001	1.062	0.165	6.44	<0.0001
Random effects (σ)	SI, Site	4.66				1.05
	SI, Plot (Site)	1.16				1.48
	Residual	0.611				0.787

) showed that the asymptote of H. macrocarpa (47 m) was greater than the asymptote of H. lusitanica (39 m). Berrill [32], using more limited data, also found that the common-asymptote Richards model performed best but was unable to distinguish differences between species.

The models can be used to predict MTH (m) at age t (years) from an initial measurement MTH₀ at age t₀, as shown by Equation (14). By setting MTH₀ to SI and t₀ to 30, Equation (14) predicts MTH for any age t from SI. Figure 2 shows the predictions using the initial measurement in each plot plotted against age along with the measured values.

$$MTH=0.3+\left(a-0.3\right){\left(1-{\left(1-{\left(\left(MT{H}_0-0.3\right)/\left(a-0.3\right)\right)}^{1/c}\right)}^{t/t_0}\right)}^c$$

(14)

By inverting Equation (14), SI can be estimated from a MTH measurement at age t years as follows:

$$SI=0.3+\left(a-0.3\right){\left(1-{\left(1-{\left(\left(MTH-0.3\right)/\left(a-0.3\right)\right)}^{1/c}\right)}^{t_0/t}\right)}^c$$

(15)

In addition, t_1.4, the age when a stand achieves a MTH of 1.4 m (breast height) can be calculated using:

$$t_{1.4}=30ln\left(1-{\left(1.1/\left(a-0.3\right)\right)}^{1/c}\right)/ln\left(1-{\left(\left(SI-0.3\right)/\left(a-0.3\right)\right)}^{1/c}\right)$$

(16)

3.2. DBH/Age Model

Table 5. Akaike Information Criterion (AIC) of nonlinear mixed models for predicting DBH from breast height age.

Model	Form	AIC
Richards	Anamorphic	9925.4
	Common-asymptote	10,137.6
Hossfeld IV	Anamorphic	10,482.2
	Common-asymptote	10,173.6
Korf	Anamorphic	9771.3
	Common-asymptote	10,137.2
	GADA	9799.4
	Anamorphic with stand density	8930.3
	Common-asymptote with stand density	8603.3
	GADA with stand density	8727.7
	Anamorphic with stand density, separate parameters for each species	8797.8
	Common-asymptote with stand density, separate parameters for each species	8598.2
	GADA with stand density, separate parameters for each species	8649.5

shows the fits of models listed in Table 2 for predicting D_q from breast height age. Based on the AIC, the anamorphic Korf model was the best-fitting model. Including N in the model using Equation (3) provided a highly significant improvement in fit (likelihood ratio test, ${\chi }_4^2$ = 848.9, p < 0.0001). However, when N was incorporated into the model, the common-asymptote form of the Korf model gave a superior fit than the anamorphic form. A further significant improvement in fit was achieved by fitting this model with separate parameters for each species (${\chi }_{10}^2$ = 25.2, p = 0.0050).

The final model used for predicting D_q is given by Equation (17) with the parameter estimates shown in

Table 6. Parameter estimates of common-asymptote Korf DBH model with stand density adjustment (Equation (17)) fitted using the model described by Equation (3). See description of Equation (3) for parameter definitions.

Parameter		H. lusitanica				H. macrocarpa
		Estimate	Std. Error	t-Value	p-Value	Estimate	Std. Error	t-Value	p-Value
Fixed effects	${\mu }_{D_{30}}$	42.64	0.94	45.4	<0.0001	41.32	0.60	68.9	<0.0001
	a	83.28	2.42	34.4	<0.0001	83.14	2.94	28.3	<0.0001
	c	0.6974	0.0211	33.1	<0.0001	0.7100	0.0264	26.9	<0.0001
	d	0.3659	0.0124	29.5	<0.0001	0.4213	0.0083	50.8	<0.0001
	f	0.4624	0.1431	3.23	0.0017	0.2645	0.0494	5.35	<0.0001
	g1	32.02	5.65	5.67	<0.0001	20.68	4.72	4.38	<0.0001
	g2	−2.39	0.78	3.06	0.0028	−0.62	0.65	0.95	0.34
Random effects (σ)	D30, Site	7.52				5.79
	D30, Plot (Site)	1.57				2.19
	Residual	0.94				1.01

. The predictions of D_q closely corresponded to the measured values (Figure 3).

$$D_q=D\left(t_{BH}^{\prime }\right)-\left(d/f\right)\left(lnN-ln300\right)ln\left(1+exp\left(f\left(D\left(t_{BH}^{\prime }\right)-\left(g_1+g_{2}lnN\right)\right)\right)\right)$$

(17)

where $D\left(t_{BH}^{\prime }\right)=a{\left(D_{30}/a\right)}^{{\left(t_{BH}^{\prime }/\left(30-t_{1.4}\right)\right)}^{-c}}$ and $t_{BH}^{\prime }=t-t_{1.4}-\Delta t_{thin}$.

3.3. Thinning Coefficient

The thinning coefficient for cypress estimated using Equation (5) from 240 thinning events recorded in the PSP data was p = 0.700 ± 0.015 (mean ± standard error).

3.4. Mortality Function

Mortality varied with age, species and location, being higher in the North Island, especially for H. macrocarpa, and higher before age 10 years but stabilising at ages greater than this.

Table 7. Parameter estimates of k (annual mortality rate) using the Equation (7) stand density model.

Location	Age (years)	H. lusitanica				H. macrocarpa
		Estimate	Std. Error	t-Value	p-Value	Estimate	Std. Error	t-Value	p-Value
North Island	<10	0.0138	0.0013	10.6	<0.0001	0.0476	0.0078	6.1	<0.0001
	≥10	0.0117	0.0007	16.7	<0.0001	0.0177	0.0015	11.8	<0.0001
South Island	<10	0.0168	0.0046	3.65	0.0004	0.0088	0.0027	3.26	0.0015
	≥10	0.0084	0.0023	3.65	0.0004	0.0052	0.0005	10.4	<0.0001

shows the estimates of annual proportional mortality (designated by parameter k) obtained using Equation (7) for each species by island for stands younger and older than 10 years.

A relationship between Reineke’s SDI and mortality was not discernible, and the Equation (9) mortality function, which includes a term describing competition-induced mortality using SDI, showed no significant improvement in fit over Equation (7). A model of the Equation (9) form has been fitted for redwood [28] and also for radiata pine and Pseudotsuga menziesii (Mirbel) Franco (Douglas-fir) using the much more extensive data available for the latter two species (these latter two models have not been formally published but are described in unpublished reports). It can be expected that a similar type of model would also apply to cypresses, but to fit such models requires data from highly stocked plots undergoing self-thinning. Examination of the cypress database revealed that it contained few such plots. Therefore, an Equation (9) mortality model could not be derived using the currently available cypress data in New Zealand.

A comparison of the upper tail of the SDI distribution using the data used to fit the radiata pine, Douglas-fir and redwood mortality functions with the two cypress species is shown in

Table 8. Percentiles of the distribution of Reineke’s stand density index (SDI) for planted stands of various exotic softwood species in New Zealand.

Species	Percentile of SDI Distribution
	90th	95th	99th
H. lusitanica H. macrocarpa	367	425	530
	448	636	859
Redwood	763	937	1214
Douglas-fir	538	616	761
Radiata pine	332	388	495

. This indicates that the SDI upper limit is highest for redwood and lowest for radiata pine. The cypresses and Douglas-fir fall between these two extremes, suggesting that the cypress species have similar levels of carrying capacity to Douglas-fir. The unpublished Equation (9) mortality function for Douglas-fir had parameter estimates of k = 0.0026, m = 0.0314 and n = 1.41. One solution to providing a mortality function which responds to competition that could be considered until suitable data becomes available for cypresses is to fit the Equation (9) k parameter for cypresses using the Douglas-fir parameter estimates for parameters m and n. The estimates for k using this approach for the two cypress species by island and age class are shown in

Table 9. Parameter estimates of k (annual mortality rate) in the Equation (9) stand density model with m = 0.0314 and n = 1.41.

Location	Age (years)	H. lusitanica				H. macrocarpa
		Estimate	Std. Error	t-Value	p-Value	Estimate	Std. Error	t-Value	p-Value
North Island	<10	0.0115	0.0013	8.85	<0.0001	0.0453	0.0078	5.81	<0.0001
	≥10	0.0049	0.0007	7.00	<0.0001	0.0103	0.0014	7.36	<0.0001
South Island	<10	0.0165	0.0048	3.44	0.0009	0.0077	0.0028	2.75	0.0071
	≥10	0.0054	0.0022	2.45	0.016	0.0000	0.0000	-	-

.

3.5. Stand-Level Volume Function and Model for Predicting MTH from Mean Height

The parameter estimates for the stand-level volume function Equation (10) are given in

Table 10. Parameter estimates for the stand-level volume function (Equation (10)).

Parameter	Estimate	Std. Error	t-Value	p-Value
u	0.2944	0.0044	66.2	<0.0001
v	0.3973	0.0073	54.2	<0.0001
w	0.5866	0.0267	22.0	<0.0001

. Because the volume model explained 99.9% of the variation in ln(V), transformation bias can be considered negligible (e.g., the transformation correction ratio based on the residual standard deviation s is exp(s²/2) = 1.0004). The parameter estimates for Equation (13), which predicts MTH from mean height, are shown in

Table 11. Parameter estimates of the model for predicting MTH from mean height (Equation (13)) The root mean square error = 0.905 m on 3110 degrees of freedom.

Parameter	Estimate	Std. Error	t-Value	P-Value
r	0.1921	0.0067	28.6	<0.0001
s	−0.00114	0.00007	−17.1	<0.0001

.

3.6. Comparison of H. lusitanica and H. macrocarpa Models

Although statistically significant differences in both the MTH and DBH models were found between species, in practice the growth patterns of the two species are very similar. Figure 4 shows that, apart from some divergence in MTH growth beyond age 30 years due to the lower asymptote for H. lusitanica, the height growth patterns of both species were almost identical, while differences in DBH growth patterns between the species were even less evident.

3.7. Model Testing

The models were tested against the fitting dataset. Firstly, the change in MTH, D_q and N was predicted for each measurement increment, and the stand-level volume function was used to predict the change in volume. The predicted versus actual volume increments were compared for various stand variables as shown in

Table 12. Performance of the growth models against various stand and site factors. For each category, actual volume periodic annual increment (PAI) was compared with its prediction, and the mean difference tested using a t-test. Also shown are the number of observations (No. obs.) used in each category. Equations (9), (14) and (17) were used to predict N, MTH and D_q, respectively, using their values at the start of the increment as initial values, and Equation (10) was used to estimate stem volume from these metrics.

Factor	Level	No. Obs.	Volume PAI (m3 ha−1 year−1)
			Actual	Predicted	Difference
Overall		1406	11.1	10.8	−0.3
Species	H. lusitanica	770	11.1	11.4	0.4
	H. macrocarpa	636	11.1	10.1	−1.0 **
Age (years)	<20	980	11.3	11.0	−0.3
	20–40	352	10.2	10.8	0.6 *
	>40	74	13.3	8.9	−4.4 **
Stand density (stems ha−1)	<500	576	10.8	10.0	−0.7 **
	500–800	350	10.6	11.3	0.8 **
	>800	480	11.7	11.1	−0.6 *
SI (m)	<22	367	8.8	8.5	−0.3
	22–28	662	11.1	10.6	−0.5
	>28	320	13.7	14.0	0.2
300 Index (m3 ha−1 year−1)	<10	350	8.6	8.8	0.2
	10–20	741	11.4	11.2	−0.2
	>20	258	13.6	12.6	−1.0

* Significantly different from zero (t test, α = 0.05), ** significantly different from zero (t-test, α = 0.01).

. Other than a tendency for the model to under-predict growth in stands older than 40 years, which were mostly H. macrocarpa stands at lower stand densities, the models showed little apparent bias across the tested stand and site factors. The under-prediction of growth at older ages appeared to result from a tendency to over-predict mortality in older stands.

An additional check of model performance was undertaken through plotting the actual stand dimensions for the last measurement in each plot against the predictions projected from the first measurement in the plot. This analysis was carried out for all plots with multiple repeated measurements. Predictions were made using the new cypress models and the earlier Berrill model [32] (Figure 5). Overall, the new models showed little apparent bias between the predictions and actual values for all stand dimensions, while the Berrill model showed a general tendency to over-predict all the stand variables, especially DBH. However, the R² values of the linear regressions of the actual vs. predicted values were only modestly higher for the new models compared with the earlier model. Because the new MTH model is of the same form as the Berrill model although with refitted parameters, it is not surprising that it provided only a minor gain in R² from 0.89 to 0.90. The improvement in R² for the DBH model (0.91 vs. 0.88) was greater, although it suggested that the DBH model form used in the new growth models provided only a modest gain in performance compared with DBH predicted using the Berrill BA projection model form.

4. Discussion

The growth models for H. lusitanica and C macrocarpa described in this paper have similar model forms to previously developed radiata pine [27] and redwood [28] growth models. These models predict DBH as a function of a starting DBH and age, projected age and stand density. Stand-level empirical growth models more commonly predict BA rather than DBH, often using simple BA projection models, which predict BA as a function of a starting BA and age, and a projected age. Models of this type were introduced by Clutter [56], who developed a BA projection model for loblolly pine using a special case of the common-asymptote Korf function. This model accounts for site by using the starting age and BA to adjust the slope parameter. Recent examples of this general approach include the use of a variety of sigmoid growth functions using advanced polymorphic formulations [57,58,59]. However, the BA projection function used in the cypress growth model of Berrill [32] is of similar form to the original Clutter model [56], it being a common-asymptote Korf function. Generally, these BA projection models do not explicitly include stand density in their formulation, although they sometimes use separate parameter estimates for different crop types such as thinned and unthinned stands [60,61].

The need for including stand density in the formulation of DBH models, such as those developed in the current study, is evident from Figure 4c, which illustrates how the DBH growth rate slowed more rapidly at higher stand densities. However, it is less obvious that there is a need to include stand density in BA projection models. At a young age, BA must be proportional to stand density, but the difference in the BA growth rate between low- and high-density stands narrows over time due to competition. A similar growth pattern may apply to stands of the same density growing on sites of different fertility. It is therefore possible that the effects of both stand density and site can be modelled adequately by a BA projection model based on a non-anamorphic growth function such as a common-asymptote model.

We can test whether a simple BA projection model for cypress is likely to perform adequately regardless of stand density by using the DBH models developed in the current study to project future BA for two stands at different stand densities but with a common starting BA at age 10 years. These could represent a fertile site at a lower stand density and a less fertile site at a higher stand density. The BA growth trends predicted by the cypress DBH growth models were very similar for both such stands (Figure 6). This suggests that for these two cypress species, simple BA projection models should perform well, and explains why the new models only provided a modest improvement in predicting growth from a starting value compared with the earlier BA projection model of Berrill [32]. However, although the hypothesis that future BA can be predicted solely as a function of initial BA and future age regardless of stand density may hold for these cypress species, at least for the example stands shown in Figure 6, it may not hold generally for other species. Figure 6 shows the predictions using the radiata pine and redwood growth models, which suggested that BA growth from a common starting value differs with stand density for radiata pine and especially redwood. This means that BA or DBH models that include stand density in their formulation should perform better than simple BA projection models for these species.

Because for cypresses the Berrill BA projection model performed quite well at projecting tree growth from a starting value, it might seem that the benefits provided by the new models insufficiently compensate for their greater complexity. However, we believe that the new models provide several important benefits which would justify their use, even if their performance at predicting BA was no better than that of a simple BA projection model. Firstly, once the SI and 300 Index are determined for a site, the new models can easily be used to predict tree size and growth for any stand density or thinning regime. Although BA projection models can be used to predict tree growth over a range of stand densities by applying simulated thinnings, there must be some doubt over the reliability of this approach. The mean DBH following thinning is affected by the thinning selection effect, and a thinned stand is not necessarily equivalent to a stand established originally at the post-thin stand density. Stand density is one of the few stand-level variables that can be directly controlled by forest managers, both by initial spacing and subsequent thinning. However, determining the optimum stand density of a stand is a complex process involving biological, technological and economic factors and requires the balancing of various constraints such as achieving a minimum tree size while maximising stand volume and maintaining stand stability. To achieve all this, models that accurately predict the effects of stand density on tree size and growth are essential. Models that inadequately predict stand density effects can lead to erroneous conclusions concerning optimum stand density. In this respect, the new models provide a significant advantage over simple BA projection models.

Another important advantage of the new models is their use of the 300 Index to express site productivity. Traditionally, foresters have used SI to measure site quality. However, the potential of a site to produce stem wood volume is only approximated by SI, which is a measure of height growth. Differential responses of height and diameter growth to key environmental factors such as wind, air temperature, water availability and nutrition have previously been observed, meaning that volume and height are only moderately correlated. Therefore, the 300 Index, which provides a direct measure of stem volume growth at a reference age and stand density, is superior to SI as a measure of site quality. In practice, BA projection models such as the Berrill cypress model account for site quality on BA growth using the BA starting value. This presents problems for stands with no available plot data or forests on new sites. Berrill provides an initial BA model for use in such circumstances [32], but this model estimates BA at a specified age and stand density only for an average site. With the new models, predictions sensitive to site variation can be made once an estimate of the 300 Index for the site is determined based on knowledge of similar sites or using a spatial productivity surface.

The new 300 Index models provide a robust hybrid framework for linking predictions to environmental determinants of growth. The thinning history and a measurement of age, N, MTH and BA or DBH at one point in time can be used by the model to generate the 300 Index and SI from plot data. When this process is repeated across many plots spanning wide environmental gradients, such as the dataset used here, a dataset of the 300 Index and SI can be assembled. Environmental covariates for each plot location can be extracted from surfaces describing key climatic, edaphic and landform features and used to construct models predicting the 300 Index and SI. Spatial predictions of these two productivity metrics can then be made from these models using the environmental surfaces as input data [15,30,31].

The utilisation of these productivity surfaces increases the flexibility of the growth models. Users do not require access to plot data to apply the growth models to particular sites and can fully parameterise the models from the values of the 300 Index and SI extracted from the surfaces. This is particularly useful for growers who are interested in assessing the productivity potential of new land under a range of silvicultural scenarios. Within New Zealand, this approach has been widely adopted by forest growers for modelling the growth and yield of radiata pine and redwood [62,63,64,65,66]. National productivity surfaces of the 300 Index and SI have been created for both species [21,30,65], which allow the relatively precise parameterisation of the two growth and yield models [28] throughout New Zealand in locations where plot data is not available. Using this approach, analyses have shown that both the 300 Index and SI can be predicted with high precision for both cypress species and the future utilisation of such productivity surfaces will greatly enhance generality of the described growth models.

Although generality is greatly improved through this approach, the growth models do not lose any of the flexibility and precision of traditional empirical models. The models can be parameterised in two ways to make predictions of growth and yield. As described above, the 300 Index and SI can be used where no plot data is available. Alternatively, the models can be fitted to stand-level metrics obtained at one point in time (age, N, MTH, DBH or BA) along with thinning information. Following this site characterisation, the metrics of stand growth can be predicted for any age and management regime including simulations covering a wide variation in stand density and in thinning number and intensity.

The empirical growth models described in this paper are based on an adequate dataset of growth measurements covering a wide range of ages, stand densities and sites across New Zealand. The models are therefore robust and can be used with confidence for predicting growth and yield of the two cypress species for New Zealand sites. The main model limitation concerns the mortality function. Competition-induced mortality is poorly modelled due to a lack of data from highly stocked stands undergoing self-thinning. The collection of data from such stands would allow an improved mortality function to be derived.

Growth and yield for radiata pine, redwood and the two cypress spp. in New Zealand have now been modelled using the method described here. Using a standardised approach based on a reference regime considerably simplifies site-species matching in two ways. Spatial comparisons of the 300 Index (i.e., volume MAI at age 30 for a reference regime of 300 stems ha⁻¹) can be made between species, which is useful for high-level comparisons of volume under clearwood regimes when the target plantation age is about 30 years. Secondly, the models described here can be used to predict the volume through time of these four species over any desired rotation length for any combination of site productivity or stand density. This provides an important refinement on the spatial comparisons described above as it allows interactions between stand density, species and age to be identified.

Growth trajectories of the different species for average North Island sites predicted using the growth models are shown in Figure 7, noting that these North Island average predictions somewhat favour radiata pine due to its much greater tolerance of cooler upland sites than the other species [15,66]. The predictions show that MTH growth for radiata pine is initially faster than the other species, although it slows considerably beyond age 30 years. The cypresses follow similar MTH growth patterns to radiata pine but scaled downwards, being 23% lower for H. macrocarpa and 31% lower for H. lusitanica at age 30 years. However, the height growth of the cypresses slows more gradually than radiata pine, and the MTH CAI of both species actually passes that of radiata pine after age 35 years for H. macrocarpa and 45 years for H. lusitanica. The more shade-tolerant redwood displays quite a different pattern of height growth compared with the other three species. The height growth of this species is slower initially but slows much more gradually with age. Redwood MTH overtakes cypress at age 15 years, while the MTH CAI passes that of radiata pine by age 30 years. At age 50 years, the redwood MTH CAI is 0.52 m year⁻¹ compared with 0.25, 0.34 and 0.20 m year⁻¹ for H. lusitanica, H. macrocarpa and radiata pine, respectively.

DBH growth for radiata pine is initially very fast but slows considerably once competition intensifies after age 10 years. DBH growth of the two cypress species is almost identical and shows a similar pattern to radiata pine though scaled downwards, with all three species slowing markedly after age 10 years. At age 30 years, the DBH of both cypress species is 17% lower than radiata pine. Redwood DBH displays a different growth pattern to the other three species, being initially slower but, as with MTH, slowing with age more gradually. Redwood DBH passes cypress at age 10 years and radiata pine at age 21 years.

Cypresses’ slower height and diameter growth produce correspondingly less volume than radiata pine, being 48% lower for H. macrocarpa and 52% lower for C lusitanica at age 30 years. The stem volume of redwood is initially lowest but passes the cypresses at age 13 years and overtakes radiata pine at age 38 years.

The pattern of stem volume vs. stem density is shown in Figure 7d. The increase in stem volume with increasing stem density is proportionately lower for radiata pine than the other species. Doubling the initial stem density from 400 to 800 stems ha⁻¹ produces an increase in stem volume of only 21% for radiata pine compared with 29% for redwood, 27% for H. macrocarpa and 33% for H. lusitanica.

Hesperocyparis macrocarpa is very susceptible to cypress canker (caused by Seiridium cupressi (Cooke and Ellis) Sutton and S. cardinale (Wagener) Sutton and Gibson), which affects branch tips, destroys stem cambium and can result in fluting and occasional tree death [36,37,38]. Observations have indicated that H. macrocarpa has a markedly higher susceptibility to cypress canker than H. lusitanica and as a consequence is more suited to colder areas such as New Zealand’s South Island where canker is less severe [36]. The results presented in this study strongly supported these observations and showed that mortality was markedly higher for H. macrocarpa than H. lusitanica within the North Island. As the disease occurs in all age classes [39,67], the decline in mortality rates with age that has been anecdotally observed most likely reflects the removal of diseased trees during the operational thinning of stands [66]. Consistent with these observations, mortality rates within the North Island dropped markedly for both species in stands greater than 10 years of age, but the mortality rates for H. macrocarpa were still double those of H. lusitanica over this age range. It is interesting to note that mortality rates within the South Island were markedly lower for H. macrocarpa than H. lusitanica, which demonstrates the strong influence of air temperature on the disease development. Cypress canker has been found to be under low to moderate genetic control within trials of 5–10-year-old trees in New Zealand, suggesting that targeted selection may improve this trait in both H. lusitanica [38] and H. macrocarpa [67].

5. Conclusions

The H. lusitanica and H. macrocarpa growth models described in this paper will be of great value to growers for regime evaluation and prediction of yield. The model predictions of key stand variables closely matched measurements. These models use the 300 Index, an index of volume productivity, which allows more precise estimation of site productivity than empirical growth models based only on SI, providing an improved framework for understanding and quantifying site productivity for these two cypress species. The height and diameter growth functions for the two species were very similar, except that the height asymptote was higher for H. macrocarpa than H. lusitanica. The mortality rates were higher in the North than the South Islands of New Zealand, especially for H. macrocarpa, which is adversely affected by cypress canker disease at warmer sites.