This section presents details about the CCT spacing trials and the observations that were used in this study, and the methods that were applied.

### The observations

Direct assessment of the potential density, and of the development towards that elusive state, requires densely stocked unmanaged and sufficiently large field plots that are enumerated regularly during long observation periods. A rare example of such an elaborate experiment is the *Correlated Curve Trend* (CCT) spacing study established by O’Connor ([1935]) in South Africa. The CCT experiment is a classic spacing trial designed to predict yields from plantations of various species of pine and eucalyptus for a wide range of densities, varying between extremely dense (2965 stems per ha) and free growth (124 stems per ha). Additional file 1: Table S1 shows details of the eight experiments used in this paper, and a map with their locations in South Africa.

The original objective of the CCT study was to predict yields from plantations of various species of pine and eucalypt for a wide range of planting densities. Most of the experiments were established between 1936 and 1938 and several detailed descriptions of the CCT design were published during the past 70 years (see for example Craib [1939]; Marsh [1957]; O’Connor [1960]; Burgers [1976]; Van Laar [1982]; Gadow [1987]; Bredenkamp et al. [2000]). Some relevant details of four spacing trials each for *Pinus patula* and *P. elliotti* are presented in Additional file 1: Table S1. The altitudes range from 53 m.a.s.l. (Kwambonambi) to 1400 m.a.s.l. (Nelshoogte), the mean annual temperatures from 15.9°C (Weza) to 21.8°C (Dukuduku) and the mean annual precipitation from 830 mm (Weza) to 1463mm (MacMac).

Soil depths and geology were also assessed in each of the eight experiments. With a depth of 120-150 cm the soils are deep in all experiments. Humic soils on granite are encountered in the MacMac and Nelshoogte experiments, sandy soils in Dukuduku and Kwambonambi and red apedal dystrophic soils in Entabeni on basalt. The mean height of dominant trees at age 20 (SI20) is lowest for *P. elliottii* at Dukuduku (17.0 m) and highest for *P. patula* at MacMac (24.3 m). Further details are presented in Additional file 1: Table S1.

The typical CCT experiment consists of 18 plots, covering 0.081 ha each. Nine of the 18 plots were left unthinned, the other nine were subjected to various thinning regimes. The treatment details for plots 1-8, and a map of the CCT spacing study Nelshoogte, are presented in Additional file 2: Table S2.

The unthinned experiment, known as the *Basic Series*, provides information about the growth of unthinned stands for a wide of range of planting densities. In the other, known as the *Thinned Series*, the response to various thinning regimes may be assessed. Eight nominal stand densities, ranging from 124 to 2965 stems per hectare, were established in plots 1 to 8 of the basic series. In order to avoid suppression by competing herbaceous flora, all plots were initially planted at 2965 stems per hectare and then thinned ‘in advance of competition’ until the nominal stocking was achieved (for further details of the trial design refer to Bredenkamp [1984] and Gadow and Bredenkamp [1992], p. 55 et sqq.). In this study only the results from several basic series are used.

Data from the CCT experiments are suitable for analysing tree survival in response to forest density and tree age. Additional climate and soil data have become available more recently, allowing a more detailed analysis of environmental effects.

### Methods to describe maximum density

This section introduces three specific approaches for analyzing maximum density and tree survival. We define the rate of survival as the ratio \frac{{N}_{2}}{{N}_{1}} while mortality may be expressed by \frac{{N}_{1}-{N}_{2}}{{N}_{1}}.

#### Limiting line

Populations of trees growing at high densities are subject to density-dependent mortality or *self-thinning*. For a given average tree size there is a limit to the number of trees that may co-exist. The relationship between the average tree size (increasing over time) and the number of live trees per unit area (declining over time) may be described by means of a “limiting relationship” or “limiting line”. A convenient model for estimating this relationship is the following:

{\mathrm{N}}_{\mathrm{max}}={a}_{0}{\mathrm{D}}^{{a}_{1}}

(1a)

where Nmax is the maximum number of surviving trees per ha_{,} D is the quadratic mean diameter [cm] and a_{0}, a_{1} are empirical parameters which can be estimated from fully stocked, unthinned trials. Reineke ([1933]) plotted the number of trees per unit area of “fully stocked stands” over their average diameter and concluded that the a_{1} parameter is a constant equal to -1.605 (Oliver and Larson [1996], pp 353–354; Zeide [2004]).

#### Nilson’s sparsity

In the case of a regular spatial distribution of the trees within a forest, the average distance between the trees (L) may be estimated by the square root of the number of m^{2} in a hectare divided by the number of trees per ha (N):

\mathrm{L}=\frac{100}{\sqrt{N}}

(2)

Nilson ([2006]) called this average distance between the trees “*stand sparsity”* and proposed to estimate the minimum distance among the survivors using the following relationship:

\mathrm{Lmin}=a+b\cdot D

(3)

Lmin is the stand sparsity (m), and D is the quadratic mean diameter (cm) of the trees in a stand; a and b are empirical parameters. The two variables have the same dimension and the relationship is assumed to be linear. Nilson’s sparsity is an alternative way of evaluating the limiting density of trees, and thus an alternative to the limiting line. We may use equations (2) and (3) to get:

{N}_{\mathit{max}}=\frac{10000}{{\left(a+b\cdot D\right)}^{2}}

(1b)

If this equation is fitted to the data, the results are almost identical to those obtained with equation (1a). In both 1a and 1b, the dependent and independent variables have different units (N per ha and cm).

#### Tree survival

Survival analysis is most often defined as a class of statistical methods for studying the occurrence and timing of events, such as death (e.g., Cox and Oakes [1984]). Survival analysis investigates the distribution of the non-negative random variable *T* which in our study describes the forest age (Staupendahl [2011]; Staupendahl and Zucchini [2011]). According to Klein and Moeschberger ([1997], pp. 21), the pattern of *T* can usually be characterised by several functions. The probability density function *f*(*t*) describes the frequency distribution of the points in time, in which trees die. In the case of continuously measured time, it is defined by:

f\left(t\right)=\underset{\Delta t\to 0}{\mathrm{lim}}\frac{P\left(t\le T<t+\Delta t\right)}{\Delta t},\phantom{\rule{1em}{0ex}}\mathrm{with}\phantom{\rule{0.5em}{0ex}}t\ge 0

(4)

For small **Δ** *t*, *f*(*t*)**Δ** *t* may be thought of as the approximate unconditional probability that tree death will occur at time *t*. The cumulative distribution function *F*(*t*), as the integral of the density function, gives the probability that a death has occurred by time *t*:

F\left(t\right)=P\left(T\le t\right)={\displaystyle \underset{0}{\overset{\phantom{\rule{0.25em}{0ex}}t}{\int}}f\left(x\right)dx}

(5)

The survival function *S*(*t*) is the complement of *F*(*t*) and gives the probability that a tree survives at least until time *t*:

S\left(t\right)=P\left(T>t\right)=1-F\left(t\right)

(6)

Because of its flexibility and parameter parsimony, Dickel et al. ([2010]) selected the Weibull distribution (Weibull [1951]). If *T* is Weibull distributed with scale parameter b and shape parameter c, the survival function is given by:

S\left(t\right)=\mathrm{exp}\left[-{\left(\frac{t}{b}\right)}^{c}\right]

(7)

with *t* >0. In this study, we will fit the survival function (Equation 7) to observed surviving *P. patula* and *P. elliottii* trees n years after planting and try to relate its parameters to the different planting espacements.