
The nonspatial density dependent crown depth and radius relationshipsThe density dependent equations for crown radius and crown depth use nonspatial measures of density to influence crown radius and crown depth. Density is measured across the plot as a whole, not locally (thus "nonspatial"). Crown Radius ParametersParameter name  Description 

NonSpatial Density Dep. Inst. Crown Height "a"  The "a" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Density Dep. Inst. Crown Height "b"  The "b" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Density Dep. Inst. Crown Height "c"  The "c" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Density Dep. Inst. Crown Height "d"  The "d" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Density Dep. Inst. Crown Height "e"  The "e" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Density Dep. Inst. Crown Height "f"  The "f" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Density Dep. Inst. Crown Height "g"  The "g" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Density Dep. Inst. Crown Height "h"  The "h" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Density Dep. Inst. Crown Height "i"  The "i" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Density Dep. Inst. Crown Height "j"  The "j" term in the instrumental crown depth equation, used to calculate crown radius  NonSpatial Exp. Density Dep. Crown Radius "D1"  The "D1" term  NonSpatial Exp. Density Dep. Crown Radius "a"  The "a" term  NonSpatial Exp. Density Dep. Crown Radius "b"  The "b" term  NonSpatial Exp. Density Dep. Crown Radius "c"  The "c" term  NonSpatial Exp. Density Dep. Crown Radius "d"  The "d" term  NonSpatial Exp. Density Dep. Crown Radius "e"  The "e" term  NonSpatial Exp. Density Dep. Crown Radius "f"  The "f" term 
Crown Depth ParametersParameter name  Description 

NonSpatial Density Dep. Inst. Crown Radius "a"  The "a" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Density Dep. Inst. Crown Radius "b"  The "b" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Density Dep. Inst. Crown Radius "c"  The "c" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Density Dep. Inst. Crown Radius "d"  The "d" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Density Dep. Inst. Crown Radius "e"  The "e" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Density Dep. Inst. Crown Radius "f"  The "f" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Density Dep. Inst. Crown Radius "g"  The "g" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Density Dep. Inst. Crown Radius "h"  The "h" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Density Dep. Inst. Crown Radius "i"  The "i" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Density Dep. Inst. Crown Radius "j"  The "j" term in the instrumental crown radius equation, used to calculate crown depth  NonSpatial Log. Density Dep. Crown Height "a"  The "a" term  NonSpatial Log. Density Dep. Crown Height "b"  The "b" term  NonSpatial Log. Density Dep. Crown Height "c"  The "c" term  NonSpatial Log. Density Dep. Crown Height "d"  The "d" term  NonSpatial Log. Density Dep. Crown Height "e"  The "e" term  NonSpatial Log. Density Dep. Crown Height "f"  The "f" term  NonSpatial Log. Density Dep. Crown Height "g"  The "g" term 
In addition to the use of density variables, the density dependent equations for crown width uses an estimate of crown depth as a dependent variable (and vice versa). This estimated value of crown width and crown depth (rad_{i} and ch_{i}) used in the density dependent equations come from the instrumental variable equations. Calculating the instrumental variables equations avoids "uncoupling" the crown radius  crown depth relationship. The nonspatial exponential density dependent crown radius function is: rad = D1 * DBH ^{a} * Height ^{b} * ch_{i} ^{c} * STPH ^{d} * BAPH ^{e} * BAL ^{f}where:  rad is the crown radius, in meters
 D1 is the NonSpatial Exp. Density Dep. Crown Radius "D1" parameter
 a is the NonSpatial Exp. Density Dep. Crown Radius "a" parameter
 b is the NonSpatial Exp. Density Dep. Crown Radius "b" parameter
 c is the NonSpatial Exp. Density Dep. Crown Radius "c" parameter
 d is the NonSpatial Exp. Density Dep. Crown Radius "d" parameter
 e is the NonSpatial Exp. Density Dep. Crown Radius "e" parameter
 f is the NonSpatial Exp. Density Dep. Crown Radius "f" parameter
 DBH is the tree's DBH, in cm
 Height is the tree height, in meters
 ch_{i} is the instrumental crown depth of the target tree, in meters, calculated using the function below
 STPH is number of stems per hectare of adult trees within the entire plot
 BAPH is the basal area, in m^{2} per hectare, of adult trees within the entire plot
 BAL is the sum of the basal area of all trees taller than the height of the target tree, in m^{2} per hectare
The instrumental equation for calculating ch_{i} is as follows: ch_{i} = a + b * DBH + c * Height + d * DBH ^{2} + e * Height ^{2} + f / DBH + g * STPH + h * BAPH + i * BAL + j * (Height / DBH)where:  a is the NonSpatial Density Dep. Inst. Crown Height "a" parameter
 b is the NonSpatial Density Dep. Inst. Crown Height "b" parameter
 c is the NonSpatial Density Dep. Inst. Crown Height "c" parameter
 d is the NonSpatial Density Dep. Inst. Crown Height "d" parameter
 e is the NonSpatial Density Dep. Inst. Crown Height "e" parameter
 f is the NonSpatial Density Dep. Inst. Crown Height "f" parameter
 g is the NonSpatial Density Dep. Inst. Crown Height "g" parameter
 h is the NonSpatial Density Dep. Inst. Crown Height "h" parameter
 i is the NonSpatial Density Dep. Inst. Crown Height "i" parameter
 j is the NonSpatial Density Dep. Inst. Crown Height "j" parameter
 DBH is the tree's DBH, in cm
 Height is the tree height, in meters
 STPH is number of stems per hectare of adult trees within the entire plot
 BAPH is the basal area, in m^{2} per hectare, of adult trees within the entire plot
 BAL is the sum of the basal area of all trees taller than the height of the target tree, in m^{2} per hectare
The nonspatial logistic density dependent crown depth function is: where:  ch is the crown depth, in meters
 height is the tree's height, in m
 a is the NonSpatial Log. Density Dep. Crown Height "a" parameter
 b is the NonSpatial Log. Density Dep. Crown Height "b" parameter
 c is the NonSpatial Log. Density Dep. Crown Height "c" parameter
 d is the NonSpatial Log. Density Dep. Crown Height "d" parameter
 e is the NonSpatial Log. Density Dep. Crown Height "e" parameter
 f is the NonSpatial Log. Density Dep. Crown Height "f" parameter
 g is the NonSpatial Log. Density Dep. Crown Height "g" parameter
 DBH is the tree's DBH, in cm
 rad_{i} is the instrumental crown radius of the target tree, in meters, calculated using the function below
 STPH is number of stems per hectare of adult trees within the entire plot
 BAPH is the basal area, in m^{2} per hectare, of adult trees within the entire plot
 BAL is the sum of the basal area of all trees taller than the height of the target tree, in m^{2} per hectare
The instrumental equation for calculating rad_{i} is as follows: rad_{i} = a + b * DBH + c * Height + d * DBH ^{2} + e * Height ^{2} + f / DBH + g * STPH + h * BAPH + i * BAL + j * (Height / DBH)where:  a is the NonSpatial Density Dep. Inst. Crown Radius "a" parameter
 b is the NonSpatial Density Dep. Inst. Crown Radius "b" parameter
 c is the NonSpatial Density Dep. Inst. Crown Radius "c" parameter
 d is the NonSpatial Density Dep. Inst. Crown Radius "d" parameter
 e is the NonSpatial Density Dep. Inst. Crown Radius "e" parameter
 f is the NonSpatial Density Dep. Inst. Crown Radius "f" parameter
 g is the NonSpatial Density Dep. Inst. Crown Radius "g" parameter
 h is the NonSpatial Density Dep. Inst. Crown Radius "h" parameter
 i is the NonSpatial Density Dep. Inst. Crown Radius "i" parameter
 j is the NonSpatial Density Dep. Inst. Crown Radius "j" parameter
 DBH is the DBH of the tree, in cm
 Height is the tree height, in meters
 STPH is number of stems per hectare of adult trees within the entire plot
 BAPH is the basal area, in m^{2} per hectare, of adult trees within the entire plot
 BAL is the sum of the basal area of all trees taller than the height of the target tree, in m^{2} per hectare
