### Bias in the big BAF estimator

The big BAF estimator has been described above and in Gregoire and Valentine (2008) as based on three sample means. Two of the sample means \(\hat {B}_{{\mathrm {c}}}\) and \(\hat {B}_{\mathrm {v}}\) provide design-based estimates of basal area per acre while the third \(\hat {V}_{\mathrm {v}}\) provides a design-based estimate of volume per hectare, albeit typically with a high variance. However their combination forms the big BAF estimator with a much lower variance but which is not design-unbiased. Here we provide equations giving a simple approximation for the bias as well as an exact upper bound to the bias associated with big BAF sampling.

#### Approximate bias

In Appendix Eq. A.11 we use a bias approximation formula from Seber (1982, p. 7) to derive the following approximation for the bias of the estimator \({\hat {V}_{\mathcal {B}}}\) in big BAF sampling:

$$\begin{array}{*{20}l} Bias &= \frac{V}{n} \left(\frac{\text{var}\!\left({\hat{B}_{\mathrm{v}_{s}}}\right)}{B^{2}} - \frac{\text{cov}\!\left({\hat{B}_{{\mathrm{c}}_{s}}}, {\hat{B}_{\mathrm{v}_{s}}}\right)}{B^{2}} \right. \notag\\ &\mspace{16.0mu}{}-\left.\frac{\text{cov}\!\left({\hat{V}_{\mathrm{v}_{s}}}, {\hat{B}_{\mathrm{v}_{s}}}\right)}{VB} + \frac{\text{cov}\!\left({\hat{V}_{\mathrm{v}_{s}}}, {\hat{B}_{{\mathrm{c}}_{s}}}\right)}{VB} \right) \end{array} $$

(15)

where \(\hat {B}_{\mathrm {v}_{s}} = m_{s}\mathcal {F}_{\mathrm {v}}\) and \(\hat {V}_{\mathrm {v}_{s}} = \mathcal {F}_{\mathrm {v}} \sum _{i=1}^{m_{\mathrm {v}_{s}}}\mathbb {V}_{i}\). Note that all the quantities in the bias expression above are population constants with respect to changing sample size except the sample size *n*. Thus as *n* goes to infinity the above expression for bias goes to zero. Bias approaching zero with increasing samples size on the order of \(\frac {1}{n}\) is similar to the behavior of the standard ratio estimator according to Cochran (1977, p. 160). Note that if the difference between the large basal area factor BAF_{c} and the small basal area factor BAF_{v} is small, the covariance between \(\hat {B}_{{\mathrm {c}}_{s}}\) and \(\hat {B}_{\mathrm {v}_{s}}\) approaches the variance for \(\hat {B}_{\mathrm {v}_{s}}\) so that the first two terms approach cancellation and similarly for the last two terms, so that bias will be also be lessened as the difference between basal area factors BAF_{c} and BAF_{v} becomes smaller. For a given sample size the bias will also be smaller for forests with high levels of basal area *B* than for forests with low levels of basal area. This bias expression is very similar to the bias that would be obtained from equation 11 of Palley and Horwitz (1961) for the Bell and Alexander (1957) estimator which can also be expressed as the ratio of two HPS sample means divided by a third sample mean. An important difference is that there are two point-wise sample sizes in the Bell and Alexander (1957) estimator, one being a point-wise subsample. Therefore some of the variances and covariances for the Palley and Horwitz (1961) bias formula and variance estimator of the Bell and Alexander (1957) volume estimator are based on the smaller subsample size while others are based on the total sample size.

#### Exact bias

In the Appendix an expression for the exact bias in the big BAF sampling estimator \({\hat {V}_{\mathcal {B}}}\), Eq. (A.15), is derived based on methods used by Hartley and Ross (1954) to find the exact bias of the standard ratio estimator (also see Cochran (1977, p. 162))

$$ Bias = \left(\mathrm{E}\!\left[ {{\hat{V}_{\mathcal{B}}}}\right] - V \right) = \frac{\text{cov}\!\left({\hat{B}_{{\mathrm{c}}}}, {\hat{V}_{\mathrm{v}}}\right) - \text{cov}\!\left({{\hat{V}_{\mathcal{B}}}}, {\hat{B}_{\mathrm{v}}}\right)}{B} $$

(16)

This formula also seems to indicate that the bias will tend to be smaller in stands having higher basal area. Again Eq. A.21 was derived in the Appendix following the methods of Hartley and Ross (1954) resulting in the following upper bound on the absolute relative bias in the big BAF estimator (also see Cochran (1977, p. 162)):

$$ \frac{\left|Bias\right|}{\sqrt{\text{var}\!\left({{\hat{V}_{\mathcal{B}}}}\right)}} \le \frac{1}{\sqrt{n}} \frac{\sqrt{\hat{B}_{\mathrm{v}_{s}}}}{B} $$

(17)

This formula indicates that the bias relative to the standard error of the big BAF estimator approaches zero as sample size *n* becomes large, on the order of \(\frac {1}{\sqrt {n}}\). This is also the case for the standard ratio estimator according to Cochran (1977, p. 160).

### The Delta method for big BAF variance based on three sample means

Previous approaches to variance estimation for big BAF sampling view the estimator as the product of two random variables, the count basal area per hectare and the mean volume basal area ratio. As indicated above, these approaches have assumed that the covariance between count basal area per hectare and the mean volume basal area ratio VBAR is negligible. However, if we do not wish to make that assumption, an alternative is to use the Delta method (Kendall and Stuart1977, p. 247), to approximate the variance of the big BAF estimator (12) in the form presented by Gregoire and Valentine (2008, equation 8.33) indicated above as a function of three sample means. On the basis of a Taylor series, the Delta method approximates the variance of a function of estimators of parameters \(g({\hat {\boldsymbol {\theta }}})\) which estimates *g*(*θ*) where *θ*=(*θ*_{1},*θ*_{2},…,*θ*_{n}). Now, since the population parameters are generally unknown, the unbiased estimators, \(\hat {\boldsymbol {\theta }}\) where \(\mathrm {E}\!\left [ {\hat {\theta _{i}}} \right ] = \theta _{i} \) are substituted here in the formula for the Delta method presented by Kendall and Stuart (1977, p. 247) *viz.*,

$$\begin{array}{*{20}l} \text{var}\!\left({g(\hat{\boldsymbol{\theta}})}\right) & \approx \sum_{i=1}^{n} \text{var}\!\left({\hat{\theta}_{i}}\right) g_{i}^{\prime}\!\left({\hat{\boldsymbol{\theta}}} \right)^{2} \notag \\ &\mspace{-2mu}{}+ 2\mathop{\sum\sum}_{i< j} \text{cov}\!\left({\hat{\theta}_{i}}, {\hat{\theta}_{j}} \right) g_{i}^{\prime}\!\left({\hat{\boldsymbol{\theta}}}\right) g_{j}^{\prime}\!\left({\hat{\boldsymbol{\theta}}} \right) \end{array} $$

(18)

or, assuming independence…

$$\begin{array}{*{20}l} &\approx \sum_{i=1}^{n} \text{var}\!\left({\hat{\theta}_{i}} \right) g_{i}^{\prime}\!\left({\hat{\boldsymbol{\theta}}}\right)^{2} \end{array} $$

(19)

In addition in typical applications it is necessary to estimate the variance and covariance terms. In this section we will assume without loss of generality that *A*=1. Let us define the function *g* in the formula for the Delta method with \(\hat {\theta }_{1} = \hat {B}_{{\mathrm {c}}}, \hat {\theta }_{2} = \hat {V}_{\mathrm {v}}\), and \(\hat {\theta }_{3} = \hat {B}_{\mathrm {v}}\) as follows:

$$ g\left(\hat{B}_{{\mathrm{c}}},\hat{V}_{\mathrm{v}},\hat{B}_{\mathrm{v}}\right)=\hat{B}_{{\mathrm{c}}}\left(\frac{\hat{V}_{\mathrm{v}}}{\hat{B}_{\mathrm{v}}}\right) $$

(20)

The Delta method requires the following three partial derivatives:

$$ \frac{\partial g}{\partial\hat{B}_{{\mathrm{c}}}}=\left(\frac{\hat{V}_{\mathrm{v}}}{\hat{B}_{\mathrm{v}}}\right) $$

(21)

$$ \frac{\partial g}{\partial\hat{V}_{\mathrm{v}}} = \left(\frac{\hat{B}_{{\mathrm{c}}}}{\hat{B}_{\mathrm{v}}}\right) $$

(22)

$$ \frac{\partial g}{\partial\hat{B}_{\mathrm{v}}} = -\left(\frac{\hat{V}_{\mathrm{v}} \hat{B}_{{\mathrm{c}}}}{\hat{B}_{\mathrm{v}}^{2}}\right) $$

(23)

Applying the Delta method and substituting estimates for variances, covariances and means we then have:

$$\begin{array}{*{20}l} \widehat{\text{var}}_{\delta_{1}}\!\left({{\hat{V}_{\mathcal{B}}}}\right) &= \left(\frac{\hat{V}_{\mathrm{v}}}{\hat{B}_{\mathrm{v}}}\right)^{2} \widehat{\text{var}}\!\left({\hat{B}_{{\mathrm{c}}}} \right)\notag\\ &\quad{}+\left(\frac{\hat{B}_{{\mathrm{c}}}}{\hat{B}_{\mathrm{v}}}\right)^{2} \widehat{\text{var}}\!\left({\hat{V}_{\mathrm{v}}} \right)\notag\\ &\quad{}+\left(\frac{\hat{V}_{\mathrm{v}}\hat{B}_{{\mathrm{c}}}}{\hat{B}_{\mathrm{v}}^{2}}\right)^{2} \widehat{\text{var}}\!\left({\hat{B}_{\mathrm{v}}}\right) \notag\\ &\quad{}+ 2\left(\frac{\hat{V}_{\mathrm{v}}}{\hat{B}_{\mathrm{v}}}\right)\left(\frac{\hat{B}_{{\mathrm{c}}}}{\hat{B}_{\mathrm{v}}}\right) \widehat{\text{cov}}\!\left({\hat{B}_{{\mathrm{c}}}}, {\hat{V}_{\mathrm{v}}} \right) \notag\\ &\quad{}- 2\left(\frac{\hat{V}_{\mathrm{v}} \hat{B}_{{\mathrm{c}}}}{\hat{B}_{\mathrm{v}}^{2}}\right) \left(\frac{\hat{V}_{\mathrm{v}}}{\hat{B}_{\mathrm{v}}}\right) \widehat{\text{cov}}\!\left({\hat{B}_{{\mathrm{c}}}}, {\hat{B}_{\mathrm{v}}} \right) \notag \\ &\quad{}- 2\left(\frac{\hat{V}_{\mathrm{v}} \hat{B}_{{\mathrm{c}}}}{\hat{B}_{\mathrm{v}}^{2}}\right) \left(\frac{\hat{B}_{{\mathrm{c}}}}{\hat{B}_{\mathrm{v}}}\right) \widehat{\text{cov}}\!\left({\hat{V}_{\mathrm{v}}}, {\hat{B}_{\mathrm{v}}} \right) \end{array} $$

(24)

The estimated variance of the volume per a unit area based on the large BAF angle gauge alone is

$$ \widehat{\text{var}}\!\left({\hat{V}_{\mathrm{v}}}\right) = \frac{\sum_{s=1}^{n}(\hat{V}_{\mathrm{v}_{s}}-\hat{V}_{\mathrm{v}})^{2}}{n(n-1)} = \frac{\widehat{\text{var}}\!\left({\hat{V}_{\mathrm{v}_{s}}}\right)}{n} $$

(25)

where

$$ \hat{V}_{\mathrm{v}_{s}} = \mathcal{F}_{\mathrm{v}} \sum_{i=1}^{m_{\mathrm{v}_{s}}}\mathbb{V}_{i} $$

(26)

which is the total volume at sample point *s* and

$$ \hat{V}_{\mathrm{v}}=\frac{\sum_{s=1}^{n} \hat{V}_{\mathrm{v}_{s}}}{n} $$

(27)

The estimated variance for the basal area per hectare based on the large angle gauge \(\mathcal {F}_{\mathrm {v}}\) is

$$ \widehat{\text{var}}\!\left({\hat{B}_{\mathrm{v}}}\right) =\frac{\sum^{n}_{s=1}(\hat{B}_{\mathrm{v}_{s}} - \hat{B}_{\mathrm{v}})^{2}}{n(n-1)} = \frac{\widehat{\text{var}}\!\left({{\hat{B}_{\mathrm{v}_{s}}}}\right)}{n} $$

(28)

where \(\hat {B}_{\mathrm {v}_{s}} = m_{s}\mathcal {F}_{\mathrm {v}}\) is the basal area per hectare at point *s* with the large basal area factor BAF_{v}. The estimated variance \(\widehat {\text {var}}\!\left ({\hat {B}_{{\mathrm {c}}}}\right)\) for the basal area per hectare based on the small angle gauge \(\mathcal {F}_{{\mathrm {c}}}\) is given by Eq. 8.

Now since Eq. 24 utilizes covariance terms, we present the computational formulas for these. Recall the relationship between the sample covariance and the estimated covariance between sample means based on *n* independent samples is:

$$ \text{cov}\!\left({\bar{X}}, {\bar{Y}}\right) =\frac{\text{cov}\!\left(X, Y\right) }{n} $$

(29)

Using this relationship the estimated covariance between \(\hat {B}_{{\mathrm {c}}}\) and \(\hat {V}_{\mathrm {v}}\) is:

$$\begin{array}{*{20}l} \widehat{\text{cov}}\!\left({\hat{B}_{{\mathrm{c}}}}, {\hat{V}_{\mathrm{v}}}\right) &= \frac{\sum^{n}_{s=1}(\hat{B}_{{\mathrm{c}}_{s}} - \hat{B}_{{\mathrm{c}}}) (\hat{V}_{\mathrm{v}_{s}}- \hat{V}_{\mathrm{v}})}{n(n-1)} \notag \\ &= \frac{\widehat{\text{cov}}\!\left({\hat{B}_{{\mathrm{c}}_{s}}}, {\hat{V}_{\mathrm{v}_{s}}}\right)}{n} \end{array} $$

(30)

the estimated covariance between \(\hat {B}_{\mathrm {v}}\) and \(\hat {V}_{\mathrm {v}}\) is:

$$\begin{array}{*{20}l} \widehat{\text{cov}}\!\left({\hat{B}_{\mathrm{v}}}, {\hat{V}_{\mathrm{v}}}\right) &= \frac{\sum^{n}_{s=1}(\hat{B}_{\mathrm{v}_{s}}-\hat{B}_{\mathrm{v}})(\hat{V}_{\mathrm{v}_{s}} - \hat{V}_{\mathrm{v}})}{n(n-1)} \notag \\ &= \frac{\widehat{\text{cov}}\!\left({\hat{B}_{\mathrm{v}_{s}}}, {\hat{V}_{\mathrm{v}_{s}}}\right) }{n} \end{array} $$

(31)

and the estimated covariance between \(\hat {B}_{\mathrm {v}}\) and \(\hat {B}_{{\mathrm {c}}}\) is:

$$\begin{array}{*{20}l} \widehat{\text{cov}}\!\left({\hat{B}_{\mathrm{v}}}, {\hat{B}_{{\mathrm{c}}}}\right) &= \frac{{\sum^{n}_{s=1}(\hat{B}_{\mathrm{v}_{s}}}-\hat{B}_{\mathrm{v}})(\hat{B}_{{\mathrm{c}}_{s}}-\hat{B}_{{\mathrm{c}}})}{n(n-1)} \notag \\ &= \frac{\widehat{\text{cov}}\!\left({\hat{B}_{\mathrm{v}_{s}}}, {\hat{B}_{{\mathrm{c}}_{s}}}\right)}{n} \end{array} $$

(32)

As is shown in Supplementary Materials equationsS.3–S.7 variance estimator (24) can also be derived as a special case of an estimator presented by Hansen et al. (1953, p. 512–514) for the variance of a ratio between the product of *k* random variables and the product of *p*−*k* random variables (Wolter2007, p. 233–234).

The variance equation can be simplified by noting that the two basal area estimators \(\hat {B}_{\mathrm {v}}\) and \(\hat {B}_{{\mathrm {c}}}\) have the same expected value *B* and the variance of \(\hat {B}_{{\mathrm {c}}}\) is likely to be smaller because it is based on the smaller BAF_{c} which selects more trees per point. In the original true variance approximation the coefficients multiplied by variances and covariances are functions of parameters which we must estimate when obtaining the approximate variance estimator. This justifies substitution of \(\hat {B}_{{\mathrm {c}}}\) for \(\hat {B}_{\mathrm {v}}\) in the variance formula above because they have the same expectation. Making this substitution and factoring out sample size *n*, the variance formula can be simplified to:

$$\begin{array}{*{20}l} \widehat{\text{var}}_{\delta_{2}}\!\left({{\hat{V}_{\mathcal{B}}}}\right) &= \frac{\hat{V}_{\mathrm{v}}^{2}}{n} \left(\frac{\widehat{\text{var}}\!\left({\hat{B}_{{\mathrm{c}}_{s}}}\right)}{{\hat{B}_{{\mathrm{c}}}^{2}}} + \frac{\widehat{\text{var}}\!\left({\hat{V}_{\mathrm{v}_{s}}}\right)}{\hat{V}_{\mathrm{v}}^{2}} \right. \notag\\ &\mspace{-4mu}{}+\frac{\widehat{\text{var}}\!\left({\hat{B}_{\mathrm{v}_{s}}}\right)}{\hat{B}_{{\mathrm{c}}}^{2}} + 2\frac{\widehat{\text{cov}}\!\left({\hat{B}_{{\mathrm{c}}_{s}}}, {\hat{V}_{\mathrm{v}_{s}}}\right)} {\hat{V}_{\mathrm{v}} \hat{B}_{{\mathrm{c}}}} \notag\\ &\mspace{-4mu}{}-\left. 2\frac{\widehat{\text{cov}}\!\left({\hat{B}_{\mathrm{v}_{s}}}, {\hat{B}_{{\mathrm{c}}_{s}}} \right) }{\hat{B}_{{\mathrm{c}}}^{2}} - 2\frac{\widehat{\text{cov}}\!\left({\hat{V}_{\mathrm{v}_{s}}}, {\hat{B}_{\mathrm{v}_{s}}}\right)}{\hat{V}_{\mathrm{v}} \hat{B}_{{\mathrm{c}}}^{2}} \right) \end{array} $$

(33)

We have derived this variance estimation formula under the assumption that *A*=1 so the variance estimate for an entire tract of area *A* can be obtained by multiplying by *A*^{2} or alternatively expressing \(\hat {V}_{\mathrm {v}}\) in total tract units rather than as per hectare. Note that because we have factored out a quantity of \(\frac {1}{n}\) the estimator above is a function of the sample variances and covariances among sample point HPS estimates.

The variance estimator above is very similar to equation (12) of Palley and Horwitz (1961) which they obtained for the Bell and Alexander (1957) estimator which was essentially double sampling with a ratio estimator. However an important difference is that the Bell and Alexander (1957) estimator consists of a large sample of points on which basal area counts are made and a subsample of points on which tree volumes are also determined. By contrast for big BAF sampling the volume subsample is made on every point so there is no smaller point-wise sample. Therefore some of the variances and covariances for the Palley and Horwitz (1961) variance estimator of the Bell and Alexander (1957) volume estimator must be determined on the subsample which is smaller than *n*, but for big BAF sampling all the variances and covariances have the same point-wise sample size of *n*. A consequence is that for the big BAF variance estimator we cannot further simplify the variance estimator above by utilizing the ratio of large-to-small point-wise sample size as was done by Palley and Horwitz (1961).

### Simulation trials

We used two simulated forest populations that were previously employed by Gove et al. (2020) to compare traditional and previously proposed big BAF sampling variance estimators. The sampling simulation program *sampSurf*Gove (2012) which was written in **R** (R Core Team 2021) was used to conduct the simulations. The concept of “sampling surface” (Williams 2001a, b) was used to construct the *sampSurf* simulator in which a raster tract of area *A* is tessellated into square grid cells. Trees are located on the tract and inclusion zones are established for each tree based on the sampling procedure (horizontal point sampling for these simulations). A sample point is considered to be located in the center of each grid cell. Totals for each grid cell are based on the attributes of trees whose inclusion zones contain the sample point at the grid cell center. The sampling surface is developed based on the total attributes values over all the grid cells. For our simulations square tracts were used with grid cells 1 m^{2} in size.

Nine sets of simulations were conducted using every combination of BAF pairs (\(\mathcal {F}_{\mathrm {v}}\) and \(\mathcal {F}_{{\mathrm {c}}}\)) where \(\mathcal {F}_{{\mathrm {c}}}\ \in \{3, 4, 5\}\) and \(\mathcal {F}_{\mathrm {v}}\ \in \{10, 20, 30\}\) for both forest populations. For each sampling simulation sampling surfaces were developed for total basal area and total volume using every combination of \(\mathcal {F}_{{\mathrm {c}}}\) and \(\mathcal {F}_{\mathrm {v}}\) resulting in 36 simulation surfaces. A Monte Carlo experiment was conducted for each of the 9 BAF factor combinations in which random samples of *n*=10,25,50 and 100 were drawn with 1,000 replications. Summary statistics were computed for HPS and big BAF sampling for each sample on each sampling surface. The statistics available for each BAF combination made it possible to compare the big BAF results to an HPS sample using \(\mathcal {F}_{{\mathrm {c}}}\) in which every sample tree was measured for volume (e.g., DBH and height measured).

#### Mixed northern hardwood population

The mixed northern hardwood population is the same one used by Gove et al. (2020). The population is artificially constructed but resembles what could typically be found in a mixed northern hardwoods forest. It is established on a tract having an area of *A*=3.17 ha and containing 31,684 grid cells. The tract is bounded by an external buffer 18 m wide so that the portion of the tract containing the tree population internal to the buffer has an area of 2 ha. A population of *m*=667 trees with a total basal area of 48.4 m^{2} was established within the tract boundaries. This is approximately equivalent to 333 trees ·*ha*^{−1} and a basal area of 24.2 m^{2} ·*ha*^{−1} with a stand quadratic mean diameter of \(\bar {D}_{q} = 30.3\) cm. According to northern hardwoods stocking guides by Leak et al. (2014) the stand would be in fully stocked condition. A three-parameter Weibull distribution (Bailey and Dell 1973) was used to assign tree DBHs, with location, scale and shape parameters respectively being *α*=10 cm, *γ* = 2 and *ζ* = 30 cm. Total heights for each tree in the simulated northern hardwoods stand were assigned using the all-species DBH-height equation by Fast and Ducey (2011) for northern hardwoods in New Hampshire converted to metric units. A normal random error term with mean zero and standard deviation 2.5 m was added to each height prediction. A spatial inhibition process (Venables and Ripley2002, p. 434) with an inhibition distance of 3 m was used to assign trees to spatial locations within the simulated northern hardwoods forest tract. The method of Masuyama (1953) for boundary overlap correction was used in which tree inclusion zones were allowed to overlap into the buffer region (Gregoire and Valentine2008, p. 224). Because random sample points can fall anywhere in the tract which includes the buffer region, each tree has a complete inclusion zone.

The following taper function is used within the *sampSurf* simulation (Van Deusen 1990):

$$ d(h) = D_{u} + (D_{b} - D_{u})\left(\frac{H-h}{H}\right)^{\frac{2}{r}} $$

(34)

where *D*_{u} is the top diameter at tree stem height *h*, *D*_{b} is the tree stem butt diameter and 0≤*h*≤*H* is tree height. The value of the taper parameter *r* was randomly selected for each tree from the range *r*∈[1.5,3]. With the taper function above a neiloidal form results if 0<*r*<2, a cone if *r*=2 and a paraboic form if *r*>2. The taper function for each tree was used to compute individual tree volume according to the procedures of Gove (2011a, p. 8). There was a correlation coefficient \(\rho (\mathbb {V}, b) = 0.62\) between individual tree VBAR and basal area in the simulated northern hardwoods population. Figure S.2 in the Supplementary Material for Gove et al. (2020) displays histograms of the DBH and height distributions for the simulated northern hardwoods forests.

#### Eastern white pine population

The eastern white pine (*Pinus strobus* L.) population used by Gove et al. (2020) was also used in this study. Gove et al. (2000) describes data collection for the eastern white pine based on Barr & Stroud FP-12 dendrometry over a 20-year period. These data were obtained from pure even-aged white pine forest stands in southern New Hampshire. Data processing utilized the **R***Dendrometry* package (Gove 2011a). The white pine population used for simulations consists of *m*=316 white pine trees with multiple measurements on some during the period. Trees were located within a 1 ha tract having an 18 m wide buffer and having a total area of *A*=1.85 ha in size with 18,496 grid cells. The population has a basal area of 47.2 m^{2} and a quadratic mean DBH of \(\bar {D}_{q} = 43.6\) cm. According to the Leak and Lamson (1999) white pine stocking guide, the tract is solidly in the full stocking range. The trees were originally measured in several different stands without location information. To assign trees spatial locations for the simulation stand, a spatial inhibition process having an inhibition distance of 3 m was employed similarly to the northern hardwoods stand discussed above. As with the northern hardwoods stand, Mayasuma’s method was used to correct for boundary overlap in point sampling, so that randomly located sample points were permitted to fall into the buffer strip surrounding the 1 ha white pine tract. No taper function was required for the white pine stand because dendrometry measurements were available for upper-stem taper on each tree. As described by Gove (2011b), a cubic spline was fitted to tree dendrometry measurements. Smalian’s formula (Kershaw et al. (2016, p. 241)) was used to calculate individual tree volumes. Figure S.5 in the Supplementary Material of Gove et al. (2020) displays histograms of DBH and total height distributions for the white pine population.