Asymptotic size–density relationships within self-thinning black spruce and jack pine stand-types: Parameter estimation and model reformulations

Abstract

The objective of this study was to quantify the asymptotic relationship between mean volume ( v ¯ (dm 3)) and density ( N (stems/ha)) within self-thinning (1) upland black spruce ( Picea mariana (Mill.) B.S.P.) stands (denoted PIm UL), (2) jack pine ( Pinus banksiana Lamb.) stands (denoted PNb), (3) upland black spruce–jack pine mixed stands (denoted PImPNb) and (4) lowland black spruce stands (denoted PIm LL). The dataset consisted of 789 v ¯ − N mid-point measurement pairs (number of pairs by stand-type = 175 PIm UL, 201 PNb, 203 PImPNb and 210 PIm LL) derived from 274 permanent sample plots (PSPs; number of PSPs by stand-type = 99 PIm UL, 64 PNb, 52 PImPNb and 59 PIm LL) situated throughout the central portion of the Canadian Boreal Forest Region. Based on the logarithmic model specification of the self-thinning rule ( log 10 ( v ¯ ) = α ′ 0 ( i ) + α 1 ( i ) log 10 ( N ) where α ′ 0 ( i ) and α 1( i) are intercept and slope coefficients specific to the ith stand-type, respectively), the analysis consisted of three basic steps: (1) data splitting and subsequent selection of nine asymptotic log 10 ( v ¯ ) –log 10( N) interval subsets per stand-type; (2) obtaining parametric and non-parametric (bootstrap and jackknife) parameter estimates for each interval subset employing ordinary least squares (OLS), bisector OLS (BIS) and reduced major axis (RMA) regression techniques, yielding a total of 81 sets of parameter estimates for each stand-type; (3) given (2), identifying the relationships and corresponding interval subset which exhibited the smallest parameter estimate variances by stand-type and subsequently selecting among the nine parameterization methods, the most appropriate relationship based on applicability of the statistical assumptions employed. The results indicated negligible differences among the regression methods and estimation procedures in terms of parameter estimates and associated variances. Conceptually, however, it was concluded that the BIS method combined with the bootstrap estimation procedure was the most appropriate given that (1) the BIS method implicitly acknowledges the underlying symmetrical bivariate relationship between the variables and (2) bootstrapping incorporates underlying distributional information in parameter estimation. Numerically, the resultant estimates and associated 95% confidence limits for the intercept and slope parameters were respectively: 7.288 (5.931/8.286) and −1.552 (−1.837/−1.153) for PIm UL (product moment correlation coefficient ( r) = −0.969); 6.216 (6.032/6.373) and −1.214 (−1.261/−1.161) for PNb ( r = −0.997); 6.145 (5.908/6.353) and −1.181 (−1.242/−1.111) for PImPNb ( r = −0.993); 7.433 (6.574/7.890) and −1.562 (−1.680/−1.309) for PIm LL ( r = −0.979). Hence the observed thinning exponents for the PIm UL and PIm LL stand-types were consistent with that predicted by both the geometric (−1.5) and allometric (−1.33) formulations, whereas those for the PNb and PImPNb stand-types were not. An alternative formulation of the self-thinning relationship for jack pine stands is postulated based on the frictional interaction among tree crowns.