Materials and Methods

A field experiment was conducted near San Simon, AZ (lat. 32°15′20.2″ N, long. 109°10′29.8″ W, elevation 1118 m) in an orchard block that was planted in 2011. In 2018, the studied area consisted of seven rows with 17 trees each, beginning with the 11th tree from the end of each row, in the interior of an orchard block. Six rows were ‘Wichita’, and one row was ‘Western’ (every fourth row in the orchard block). In 2019, all of the same tree rows were used as in the previous year plus two additional adjacent rows of ‘Western’ with 17 trees each were included. Thus, in the second year, the study included six rows of ‘Wichita’ and three of ‘Western’. Both scion cultivars were on open-pollinated ‘Ideal’ rootstock. The soil was Vekol loam (Fine, mixed, superactive, thermic Typic Haplargids). The orchard’s climate is arid, and it receives ≈24 cm of precipitation annually (Western Regional Climate Center, n.d.). The orchard was irrigated through a microsprinkler system, with one sprinkler halfway between adjacent trees, 24 times per year at a rate of ≈6.4 cm per irrigation (≈152 cm annually). Each sprinkler wetted a circle of ≈7.3 m in diameter. Nitrogen, phosphorus (P), and potassium (K) were applied uniformly through the fertigation system on five occasions, March through June each year, at annual rates of 213 kg·ha^–1, 50.5 kg·ha^–1, and 50.5 kg·ha^–1, respectively. Zinc application rates were determined by the grower-cooperator. Nine percent Zn-EDTA (ethylenediaminetetraacetic acid) was applied on eight occasions by fertigation at a total rate of 6.0 kg·ha^–1 of Zn in 2018 and 11.0 kg·ha^–1 of Zn in 2019. No foliar Zn applications were made to the trees during this study. In both years, 2.24 kg·ha^–1 K and 1.12 kg·ha^–1 nickel (Ni) were foliarly applied in April/May, and 4.48 kg·ha^–1 K and 2.24 kg·ha^–1 iron (Fe) were foliar applied in June. Standard commercial weed and insect control measures were conducted by the grower-cooperator.

In 2018 and 2019, foliar samples consisting of at least 20 pairs of leaflets were collected from fruiting branches of each tree in the experimental plot following a standard sampling protocol for leaflet selection (Graham, 2021). All leaflets were washed in a phosphorus-free detergent, and then rinsed in deionized water, followed by a 1% hydrochloric acid bath, and a final rinse in deionized water. The leaflets were spun dry and placed in an oven for 48 h at 65 °C and ground using a cyclone mill (UDY Cyclone Sample Mill, UDY Corporation, Fort Collins, CO). Samples were analyzed for total nutrient content (Brookside Laboratories, Inc., New Bremen, OH).

Soil samples, each consisting of four combined subsamples, were collected from near the base of each tree in the experimental plot in 2018 and 2019 to a depth of 0 to 12 inches. The soil samples were oven-dried at a temperature of 65 °C, ground and screened using a 2-mm sieve. Samples were analyzed for extractable nutrient contents, pH, and electrical conductivity (Brookside Laboratories, Inc.). Trunk diameters were measured at 75 cm above the ground surface on 28 Feb. 2019 and 14 Jan. 2020.

Leaf gas exchange was measured on middle leaflets from nonterminal, sun-lit leaves from each tree in the ‘Wichita’ rows on 20 Sept. 2019 using a portable gas exchange unit (LI-6800; LI-COR, Lincoln, NE) equipped with a red/blue LED light source (LICOR 6800-02). Photosynthetically active radiation (PAR) in the chamber was maintained at 1700 mmol·m^–2·s^–1. Light saturation of A (photosynthesis) for pecan is reached between a PAR of 1500 to 1700 mmol·m^–2·s^–1 (Anderson, 1994; Lombardini et al., 2009). Reference CO₂ concentration was kept at 410 mmol·mol^–1, near the global mean atmospheric concentration (U.S. National Oceanic and Atmospheric Administration, 2021). Once the A and g_S stabilized (typically between 30 and 60 s after the chamber was clamped onto the leaf) gas exchange data were logged for each leaf. Gas exchange measurements were taken between 0900 and 1300 hr.

Relationships between soil nutrients, CEC, and pH vs. leaf nutrients; between leaf nutrients; between leaf nutrient concentrations and tree growth; and between Pn vs. tree growth and mean leaf nutrient concentrations were assessed using Spearman’s ρ correlations. Relationships were assessed individually for each cultivar during each year of the experiment.

We used a mixed model to assess fixed variety effects for log-transformed Zn in 2019. We considered competing models with different spatial covariance structures including Gaussian, spherical and power structures accounting for autocorrelation among the trees within the same row and conducted inference using the model with the lowest small sample bias-corrected Akaike information criterion. A preliminary analysis incorporating fixed trends (linear only, and linear and quadratic) for row and tree position was fitted to confirm adequacy of the model with fixed effects for variety only.

Preliminary exploration indicated that foliar Zn had a positively skewed distribution that was resolved through log transformation. Consequently, formal inference to assess differences in varieties used log-transformed Zn. However, because composite samples require analysis on the untransformed scale (Lancaster and Keller-McNulty, 1998), sample size computations to establish the sample size required to estimate the mean within a prescribed margin of error used untransformed Zn.

Determination of the required sample size when sampling from an infinite or very large population to obtain a specified percentage of sample means falling within a predetermined relative margin of error from the true population mean was accomplished using the following Eqs. [1] and [2].

Solving the equation for Mr_α for n determines the sample size required to achieve a specified relative margin of error at the error rate α for a given cv. n = {z_1−α/2 · [σ/(Mr_α· µ)]}² = [ z_1−α/2 · (cv/Mr_α)]² [1]

Carvalho et al. (2020) and Miyamoto and Cruz (1986) used the special case of Eq. [1] with the relative sampling error (Er) in place of Mr_α and z_1−α/2 fixed at 1, essentially setting α to 0.32. Eq. [1] allows the flexibility to set α to any desired level and so allows establishing the sample size required for the sample mean to fall within the stated margin 100 · (1 – α)% of the time rather than fixing that value at 68%.

When sampling from a finite population with size N, we reduce the standard error of the sample mean by the multiplicative correction factor {[(N – n)/(N – 1)]^½}. For a given sample size, Mr_α,N will be smaller than Mr_α. We use the notation n_N to differentiate a sample size adjusted to account for the finite population size, N, from the sample size computed without regard to population size.Mr_α,_N = {z_1−α/2 · [(N − n_N)/(N − 1)]^½ · (cv/√n_N)}

Solving the revised margin of error formula for n_N produces a formula (Eq. [2]) to adjust the sample size n computed using Eq. [1] to account for the finite population size, N.n_N = (N · n)/(N − 1 + n) = [N/(N − 1 + n)] · n [2]

For n ≥ 1, [N/(N – 1 + n)] ≤ 1 and so n_N ≤ n. For a given sample size, as population size increases [N/(N – 1 + n)] approaches 1 and n_N is approximated well by n. Because sample sizes are constrained to be integer-valued, to ensure that at least 100 · (1 – α)% sample means fall within the indicated relative margin of error, the last step in the computation is to set n or n_N to the smallest integer that equals or exceeds the value produced by the formula.

Minimum acceptable orchard block-scale mean foliar Zn concentrations to avoid or minimize Zn deficiency based on the test orchard block and for a range of cv values from 0.15 to 0.40 were established using 90% confidence intervals (CI) for the fifth percentile of a lognormal distribution using only a sample mean and an assumed known cv value (see derivation that follows). The lower bound for the 90% confidence interval is the same as the one-sided lower 95% tolerance bound met by at least 95% of the trees (Hahn and Meeker, 1991). The upper bound corresponds to a one-sided upper 95% tolerance bound met by at least 5% of the trees. For samples of size 35, we computed the minimum acceptable sample or composite mean as the smallest sample mean resulting in a confidence interval lower bound greater than 15 mg·kg^–1. We also computed the largest sample mean that results in the confidence interval upper bound less than 15 mg·kg^–1 to identify the range of sample means that indicate current Zn protocols are inadequate.

If a simple random sample X_i, i = 1 to k, is from a lognormal distribution with mean ${\mu }_X$ and variance ${\sigma }_X^2$ then the random sample Y_i = ln (X_i), i = 1 to k, is from a normal distribution with mean ${\mu }_Y$ and variance ${\sigma }_Y^2\text{.\hspace{0.17em}Suppose \hspace{0.17em}\hspace{0.17em}that \hspace{0.17em}\hspace{0.17em}}\overline{X}$ = (1 / k) ∑ X_i is the sample mean. Alternatively, $\overline{X}$ might represent a single composite sample measurement if each sampled unit contributes equally to the composite sample and if measurement error is negligible. The sample size k may be n or n_N as computed using Eqs. [1] or [2], but the following applies to any sample size, k.

To derive the confidence interval for the pth percentile using a sample mean from a log-normally distributed random variable with known cv, we follow the approach for deriving a confidence interval for the mean of a lognormal distribution using a single composite sample as outlined by van der Voet (2005). We use the following:

1. Using the convention of representing the estimate of a parameter with hat notation, we estimate ${\mu }_X$ with ${\widehat{\mu }}_X=\overline{X}$.

2. For cv = $\frac{{\sigma }_X}{{\mu }_X}$  = c, a known constant, the variance of $\overline{X}$, V ($\overline{X}$) = ${\mu }_X^2(\frac{c^2}{k})$, can be estimated by substituting $\overline{X}$ in place of ${\mu }_X$.

3. For known cv = c, ${\widehat{\mu }}_Y=\hspace{0.17em}ln(\raisebox{1ex}{$\overline{X}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{(c^2\hspace{0.17em}+\hspace{0.17em}1)}$}\right.)\mathrm{and}{\widehat{\sigma }}_Y^2=ln(1+c^2)$.

4. A point estimate of the pth percentile for the normally distributed Y_i is given by ${\widehat{\mu }}_Y+z_p{\widehat{\sigma }}_Y$ where $z_p$ is the pth percentile of the standard normal distribution. So

   $$({\widehat{\mu }}_Y\hspace{0.17em}+\hspace{0.17em}{z}_p{\widehat{\sigma }}_Y)\hspace{0.17em}=\hspace{0.17em}ln(\raisebox{1ex}{$\overline{X}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{(c^2+1)}$}\right.)+\hspace{0.17em}{z}_p\sqrt{(\ln (1+c^2))}\hspace{0.17em}=\hspace{0.17em}ln(\overline{X})-\frac{1}{2}ln(c^2+1)\hspace{0.17em}+\hspace{0.17em}{z}_p\sqrt{ln(1+c^2)}.$$
5. Using the delta method (van der Voet, 2005) to find the variance of this point estimate produces

   $$\mathrm{Var}[ln(\overline{X})-\frac{1}{2}ln(c^2+1)+z_p\sqrt{ln(1+c^2)}]=\hspace{0.17em}\mathrm{Var}[ln(\overline{X})]=\left(\frac{c^2}{k}\right)$$

   Taking the square root produces the standard error.

6. A (1 – α) · 100% CI for the normally distributed variable, Y, becomes

   $$\ln (\overline{X})-\frac{1}{2}\ln (c^2+1)+z_p\sqrt{ln(1+c^2)}\pm z_{1-\frac{\alpha }{2}}(\frac{c}{\sqrt{k}}).$$

7. Exponentiate both endpoints to get a (1 – α)100% CI for the pth percentile of the log-normally distributed variable, X_i.

JMP Pro 15 software (SAS Institute, Cary, NC) was used to compute nonparametric Spearman’s ρ correlations. The mixed model was fitted using SAS version 9.4 software (SAS Institute Inc., 2016). We used an alpha value of 0.05 in all statistical tests. Minimum acceptable orchard block sample means for cvs ranging from 0.15 to 0.40, and sample sizes using Eqs. [1] and [2] for α = 0.05 and 0.10, relative margins of error ranging from 0.05 to 0.15, cv = 0.30, and N = 100, 250, 500, 1000, 2500, and 5000 were computed using SAS version 9.4 software (SAS Institute Inc., 2016).

Results

We measured concentrations of foliar Zn and other essential nutrients in composite samples from each individual tree. Averages of these nutrient concentrations and associated variabilities are shown in Table 1. Leaf Zn, along with B (boron) and Mn (manganese), had the greatest cv in each cultivar and for each year. Zinc concentration variability was highest in ‘Wichita’ in 2019 (cv = 0.255) and lowest in ‘Western’ during 2018 (cv = 0.186) with the highest cv of 0.30 observed when combining the varieties in 2019. The least variability was observed in leaf N.

Table 1.

Leaf nutrient concentration means (in percentage or mg·kg^–1), sd, and cv for ‘Wichita’ and ‘Western’ pecan trees sampled individually in late July and early August of 2018 and 2019.

View Table

Table 2.

Sample size (number of trees contributing to a composite sample) required to attain error rates of 0.05 or 0.10 for specific relative margins of error for various sized orchard blocks (cv = 0.30).

View Table

Zinc status of pecan trees varied both by year and by cultivar (Fig. 1). ‘Western’ trees contained more foliar Zn than ‘Wichita’ trees. In 2018 and 2019, 7.8% and 27.4% of ‘Wichita’ trees, respectively, had foliar Zn levels below 15 mg·kg^–1. In ‘Western’, 5.9% of trees had less than 15 mg·kg^–1 Zn in 2018 and in 2019, all ‘Western’ trees had foliar Zn levels greater than 15 mg·kg^–1. In ‘Wichita’, 2.9% and 1.0% of trees had foliar Zn levels greater than 30 mg·kg^–1 vs. 5.9% and 21.6% of ‘Western’ trees in 2018 and 2019, respectively.

Distribution of Zn concentrations (mg·kg⁻¹) of ‘Wichita’ and ‘Western’ pecan trees in 2018 and 2019. Those in the lowest category (<15 mg·kg^–1) are considered to be Zn deficient.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

View Full Size

Distribution of Zn concentrations (mg·kg⁻¹) of ‘Wichita’ and ‘Western’ pecan trees in 2018 and 2019. Those in the lowest category (<15 mg·kg^–1) are considered to be Zn deficient.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

Distribution of Zn concentrations (mg·kg⁻¹) of ‘Wichita’ and ‘Western’ pecan trees in 2018 and 2019. Those in the lowest category (<15 mg·kg^–1) are considered to be Zn deficient.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

Individual tree Zn status is shown in Figs. 2 and 3 for 2018 and 2019, respectively. Higher average leaf Zn concentrations were found in ‘Western’ rows vs. those of ‘Wichita’. ‘Western’ mean leaf Zn concentrations were 22.75 mg·kg^–1 in 2018 and 25.12 mg·kg^–1 in 2019 vs. 19.44 mg·kg^–1 in 2018 and 17.76 mg·kg^–1 in 2019 in ‘Wichita’ (Table 1).

Representation of experimental plot in 2018. Size of circles correspond to leaf Zn concentration of individual trees (larger circles = higher leaf Zn concentrations). Actual leaf Zn concentrations in mg·kg⁻¹ are given next to circles.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

View Full Size

Representation of experimental plot in 2018. Size of circles correspond to leaf Zn concentration of individual trees (larger circles = higher leaf Zn concentrations). Actual leaf Zn concentrations in mg·kg⁻¹ are given next to circles.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

Representation of experimental plot in 2018. Size of circles correspond to leaf Zn concentration of individual trees (larger circles = higher leaf Zn concentrations). Actual leaf Zn concentrations in mg·kg⁻¹ are given next to circles.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

Representation of experimental plot in 2019. Size of circles correspond to leaf Zn concentration of individual trees (larger circles = higher leaf Zn concentrations). Actual leaf Zn concentrations in mg·kg^–1 are given next to circles. Note that three rows of ‘Western’ trees were sampled in 2019 vs. just one row in 2018.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

View Full Size

Representation of experimental plot in 2019. Size of circles correspond to leaf Zn concentration of individual trees (larger circles = higher leaf Zn concentrations). Actual leaf Zn concentrations in mg·kg^–1 are given next to circles. Note that three rows of ‘Western’ trees were sampled in 2019 vs. just one row in 2018.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

Representation of experimental plot in 2019. Size of circles correspond to leaf Zn concentration of individual trees (larger circles = higher leaf Zn concentrations). Actual leaf Zn concentrations in mg·kg^–1 are given next to circles. Note that three rows of ‘Western’ trees were sampled in 2019 vs. just one row in 2018.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

In 2019, the best-fitting mixed model for log-transformed Zn incorporated a Gaussian + nugget spatial structure to the errors within rows. This fitted error structure accounted for the serial autocorrelation observed among tree positions within rows (Figs. 2 and 3). A preliminary model fitted incorporating linear or linear–quadratic row or tree position effects did not suggest the existence of systematic trends across the orchard block, so we based inference on the model fitting only variety fixed effects. Varieties differed (P = 0.0034), with ‘Western’ having 0.36 (± 0.08) log (mg·kg^–1) higher mean log Zn than ‘Wichita’ [‘Western’ mean 3.20 (± 0.07) vs. ‘Wichita’ 2.84 (± 0.05) log (mg·kg^–1)]. A 95% back transformed confidence interval for the difference estimates that the median Zn in ‘Western’ was between 18% and 75% higher than the median for ‘Wichita’. The model’s fitted spatial structure estimates a correlation for log Zn of 0.47 between consecutive trees within a row, 0.41 between trees two positions apart, and for trees six positions apart, a correlation of only 0.09 with the correlation quickly decreasing to negligible for trees spaced farther apart.

We assessed associations between soil and leaf nutrient levels to determine the extent to which variability of soil nutrient levels could explain the observed variability of leaf composition. A consistent positive soil to leaf correlation was found only between soil Cu (copper) and leaf S (sulfur) in ‘Wichita’. In ‘Western’ the sole consistent (positive) soil to leaf correlation was between soil Zn and leaf S (data not shown). No consistent correlations were found between soil cation exchange capacity or pH and any leaf nutrients. Increase in trunk diameter from Feb. 2019 to Jan. 2020, and 2019 leaf Zn concentrations were positively correlated (R² = 0.236) (data not shown). Other leaf nutrient concentrations correlated with tree growth included a positive correlation with N and negative correlations with K and Cu (data not shown). Measured rates of photosynthesis were not significantly correlated with any foliar nutrient concentrations, nor with growth of trunk diameter. Nonparametric Spearman’s ρ correlations showed that leaf Zn was positively correlated with leaf Cu (ρ = 0.356 and 0.388 for ‘Wichita’ and 0.716 and 0.481 for ‘Western’ in 2018 and 2019, respectively).

When the cv = 0.30 as was observed in the 2019 data combining both varieties, for 95% (i.e., an error rate = 0.05) of random samples to achieve a relative margin of error magnitude of 0.10 or less requires a sample size of 35 trees for orchard blocks with an infinite number of trees (Table 2). Fewer trees are required to achieve either a larger relative margin of error or if an error rate of 0.10 is acceptable. If having only 90% of random samples fall within the stated margin is acceptable, then a relative margin of error of 0.10 requires sampling only 25 trees. For an infinite orchard block size, achieving the smaller relative margin of error of 0.05 with 95% of samples requires a much larger sample size of 139 trees.

For a stated relative margin of error and error rate, as orchard block size decreases, the required sample size may decrease. This is particularly obvious for a relative margin of error of 0.05 where for the error rate of 0.05, the required sample size for an orchard block with infinite size is 139 trees, which drops to 135 for a 5000 tree orchard block and to 59 for an orchard block with 100 trees. For larger relative margins of error, the sample size formula for an infinite population provides a good approximation for most orchard block sizes. The multiplicative adjustment in Eq. [2], [N/(N – 1 + n)], to the sample size, n, computed from Eq. [1], suggests that if n – 1 (or n) is small relative to N, then using the simpler Eq. [1] provides an adequate approximation. For example, at the relative margin of error of 0.08, the required sample size of 55 trees for infinite orchard blocks was just over 1% of 5000 and applying the finite population multiplicative adjustment results in a minimal adjustment to the sample size of one less tree. At the opposite extreme, for a relative margin of error of 0.05 with orchard blocks of 100 trees, the sample size for an infinite orchard block is 139, so that n – 1 is 138% of the orchard block size and applying the finite population multiplicative adjustment reduces the required sample size by more than half to 59 trees.

At a cv of 0.30, for a composite sample based on 35 trees, the minimum acceptable orchard block sample mean was 27.59 mg·kg^–1 (Fig. 4). That is, a mean of 27.59 mg·kg^–1 or higher suggests the current Zn fertigation protocols are adequate. A sample mean of 23.35 mg·kg^–1 or lower suggests current Zn application protocols are not adequate (Fig. 4). Sample means between those values result in 90% CIs for the fifth percentile that include 15 mg·kg^–1 as well as values greater and less than 15 mg·kg^–1 and are inconclusive.

Minimum (dashed line) acceptable and maximum (solid line) unacceptable sample means vs. the coefficient of variation, based on a 90% confidence interval for the fifth percentile, a threshold of 15 mg·kg^–1, and a composite sample comprising 35 trees. A sample mean below the solid line suggests the Zn level is inadequate, with more than 5% of trees having suboptimal Zn. A sample mean above the dashed line suggests the Zn level is adequate with fewer than 5% of trees with suboptimal Zn.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

View Full Size

Minimum (dashed line) acceptable and maximum (solid line) unacceptable sample means vs. the coefficient of variation, based on a 90% confidence interval for the fifth percentile, a threshold of 15 mg·kg^–1, and a composite sample comprising 35 trees. A sample mean below the solid line suggests the Zn level is inadequate, with more than 5% of trees having suboptimal Zn. A sample mean above the dashed line suggests the Zn level is adequate with fewer than 5% of trees with suboptimal Zn.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

Minimum (dashed line) acceptable and maximum (solid line) unacceptable sample means vs. the coefficient of variation, based on a 90% confidence interval for the fifth percentile, a threshold of 15 mg·kg^–1, and a composite sample comprising 35 trees. A sample mean below the solid line suggests the Zn level is inadequate, with more than 5% of trees having suboptimal Zn. A sample mean above the dashed line suggests the Zn level is adequate with fewer than 5% of trees with suboptimal Zn.

Citation: HortScience 57, 4; 10.21273/HORTSCI16463-21

• Download Figure

The computation of a minimum acceptable orchard block sample mean is sensitive to both the assumptions that individual tree Zn levels are log-normally distributed and the assumed cv. The minimum acceptable mean increases steadily with increasing cv. For cv = 0.15, the minimum acceptable orchard block sample mean is only 20.21 mg·kg^–1; for cv = 0.40, it has increased to 34.03 mg·kg^–1.

Discussion

The observed mean leaf Zn concentration of ‘Wichita’ was lower than that of ‘Western’ during both years. Wakeling et al. (2001) and Walworth et al. (2017) also reported that leaf Zn concentrations were slightly higher in ‘Western’ than ‘Wichita’. Heerema (2013) and Herrera (2005) noted that when grown in high pH, calcareous soils ‘Wichita’ pecans are more susceptible to Zn deficiency than ‘Western’ trees.

The majority of ‘Wichita’ leaf Zn concentrations were between 15 and 25 mg·kg^–1 both years (Fig. 1). In 2018, 7.8% of ‘Wichita’ trees fell below the minimum threshold of 15 mg·kg^–1 vs. 27.5% that fell below this value in 2019. ‘Western’ had a higher mean leaf Zn concentration in 2019 than 2018. In 2018, 5.9% of the trees had less than 15 mg·kg^–1 of Zn, whereas none fell into this category in 2019. In 2018, just 23.5% of ‘Western’ trees contained more than 25 mg·kg^–1. This fraction more than doubled to 49.1% in 2019.

The orchard block was uniformly fertigated, and weed and pest management was well maintained. Both Zn and N were applied entirely by fertigation through the permanent microsprinkler system, so uniformity of distribution for fertilizers of these nutrients was likely similar, suggesting that fertilizer distribution was not responsible for the higher variability of foliar Zn than N. No consistent relationships between soil nutrients and leaf Zn concentrations were found, suggesting that varying soil nutrient availability was not a major source of leaf Zn concentration variability. Neither were consistent relationships found between leaf Zn concentration and soil salinity or pH.

End rows and end trees were not included in this study, and sampled trees were in the interior of the orchard block, so exposure to sunlight was relatively uniform and air flow throughout the trees was consistent. The land in the orchard block was level and visible soil characteristics were uniform. We did not observe leaf Zn concentrations effects due to tree position within rows in either year, nor were there differences between rows of each cultivar, suggesting that tree-to-tree leaf Zn concentration variability was not due primarily to position in the field but was likely related to cultivar differences and individual genetics of the rootstocks. The mixed model analysis of the 2019 data also provided no evidence of position or row trends across the orchard, but this analysis detected cultivar differences. The best fitting mixed model also implied a low degree of spatial autocorrelation with relatively low correlations between trees within a few positions of one another within the same row. All trees were grafted onto open-pollinated rootstock of ‘Ideal’ maternal parentage. Variability among individual rootstocks is a likely source of the observed variability within each scion cultivar. This is comparable with the findings of Surucu et al. (2020) who found variable Zn uptake among pistachios grafted to open-pollinated rootstock. In contrast, two studies conducted in different locations with 18 pecan cultivars found no significant difference in foliar Zn levels among any of the cultivars, suggesting limited genetic influence on variability in Zn uptake (Sparks and Madden, 1977; Worley and Mullinix, 1993).

Variability in relation to the mean, reflected in the cv, is the primary factor affecting calculation of the sample size needed to obtain results that approach the true population mean of the orchard within an acceptable relative margin of error. This relationship can be seen in the study by Miyamoto and Cruz (1986) who found that as the cv increased for mapping units of an orchard in the El Paso Valley, the sample size required to obtain a soil salinity mean within 15% of the true mean increased. They used a modified version of Eq. [2] to determine sample sizes.

A sample size of 35 trees was determined from the 2019 combined data of both cultivars to be necessary to achieve a relative margin of error of 10% and 95% confidence in an orchard block with more than 1000 trees using Eq. [2]. This sample size is considerably larger than the 10 trees recommended by O’Barr and McBride (1980) or Pyzner (n.d.) for pecan and slightly larger than the 18 to 28 trees suggested by Brown et al. (n.d.) for almonds.

Commercially, sample size selection will result from a balance of cost and practicality of sampling vs. an acceptable relative margin of error. Of course, the results obtained in this study are valid strictly for the conditions of our data only but should provide a reasonable starting point for determining the requisite number of trees to sample in an orchard block.

In research, it may be desirable to intensively sample all the treated trees in a study to maximize the ability to detect tree responses. Such data are also required to assess relationships between leaf nutrient concentrations and plant performance measures such as rates of growth or photosynthesis and nut yield, for example (Heerema et al., 2014, 2017; Sherman et al., 2017), but it is challenging to measure individual trees for routine orchard block scale decision-making. To make field-scale mean leaf Zn concentration recommendations, determination of a target mean leaf nutrient concentration needs to consider variability of trees in a sampled block and the relative margin of error associated with the collected leaf sample. Individual tree mean leaf Zn concentration in unsprayed trees fertigated with Zn-EDTA needs to be maintained at a minimum of ≈15 mg·kg^–1 (slightly above the minimum concentration noted by Heerema et al., 2017 and Hu and Sparks, 1991) to ensure photosynthesis is not Zn limited.

The mean leaf Zn concentration of an orchard block sample required to ensure a sufficient Zn concentration in 95% of the trees is, in part, dependent on the level of tree-to-tree variability. As variability increases, so does the mean leaf Zn concentration required to avoid trees limited by unacceptably low Zn concentrations. On the basis of a Zn cv of 0.30 (the upper end of the variability we observed), an appropriate field-scale minimum Zn concentration to maintain at least 15 mg·kg^–1 of foliar Zn in 95% of trees is ≈27.5 mg·kg^–1.

These guidelines are based on the studied orchard block, use the greatest Zn cvs we observed, and were computed assuming tree zinc follows a log normal distribution as was observed in the studied orchard block but provide initial target values for pecan orchard blocks in general.

Even the point estimate of the fifth percentile is sensitive to the assumed cv and distribution, whereas violations of other assumptions used when converting information from the individual tree measurements into recommendations for decisions based on a single composite sample from an orchard block may primarily affect the variability of the composite sample measurement. These include assuming that measurement error contributed minimally to the variability observed among tree zinc measurements and would contribute minimally to a composite measurement. The guidelines were formed also assuming that when obtaining a composite measurement from sampled trees, each tree would be represented equally in the measurement. Sampled trees not being represented equally in the composite measurement can be caused by unequal amounts of material being taken from trees or by incomplete mixing or homogenizing of the composited sample. We did not explore whether there is a limit on the number of trees that should be included in a single composite sample to avoid inflating the variance of the composite measurement simply because of incomplete mixing. Aspects of sampling protocols and composite sample formation and measurement might require further examination. Here where we measured individual trees, each sample consisted of at least 20 pairs of leaflets to have sufficient material to characterize the status of each tree with sufficient accuracy. Protocols for subsampling selected trees as well as how many trees one can incorporate into a single composite sample without introducing too much variability from unequal representation of trees included in the composite may require additional consideration.

No comparable data have been published to determine Zn cvs in other pecan orchards, nor do we know how management practices affect the cv or the importance of cultivar. More precise orchard management guidelines require an assessment of individual orchard or orchard block variability (McCraw et al., n.d.) or else determination of tree-to-tree variability in additional pecan orchards. The minimum acceptable orchard block means computed here are particularly sensitive to the assumed cv, which is obvious in Fig. 4, but because minimum acceptable orchard block means were based on confidence intervals for percentiles, they are also sensitive to the assumed distribution (Hahn and Meeker, 1991), which should also be considered in future studies.

Additionally, although sampling protocols often recommend that cultivars within mixed blocks be sampled separately, our experience indicates that this is not a common practice in commercial orchards. This makes sense because all trees within a block are usually treated the same, regardless of cultivar. Differences we observed between ‘Western’ and ‘Wichita’ suggest that, for the purposes of evaluating efficacy of Zn management programs, perhaps only the cultivar accumulating less Zn (‘Wichita’, in our case) should be sampled. In the southwestern United States and northern Mexico, where ‘Wichita’ and ‘Western’ are the two dominant cultivars, this could be accomplished, but Zn accumulation in most cultivars has not been well characterized.

Published sampling recommendations are apparently based on experience rather than data. We present an initial assessment of sample size selection. We have also presented a databased recommendation for a foliar Zn threshold to avoid Zn deficiency in Zn-fertigated orchards.