Table 1. Site index (H40, Tveite 1977) and characteristics of the taper sample trees. Mean values of diameter and height for sample trees within each site, with min-max in parenthesis. Diameter and bark thickness at breast height. Crown height up to the lowest green branch whorl.
Site no. Site index
(H40, m) 		No. of taper trees D
(cm) 		H
(m) 		Double bark (mm) Crown height (m) County
25 21 15 27 (6–55) 23 (7–34) 14 11 Akershus
33 12 15 16 (7–26) 16 (9–23) 11 11 Akershus
37 12 15 23 (11–41) 17 (10–24) 17 9 Nord-Trøndelag
39 7 16 15 (7–24) 13 (7–17) 12 8 Nord-Trøndelag
42 20 15 23 (7–34) 20 (9–25) 11 15 Akershus
62 14 15 16 (10–32) 15 (10–20) 12 8 Nordland
V70 24 2 81 (70–92) 42 (41–43) 27 23 Hordaland
91 23 16 14 (6–24) 15 (7–22) 8 7 Akershus
139 16 16 12 (4–21) 10 (4–17) 9 4 Oppland
143 12 15 23 (11–39) 17 (10–24 18 8 Nord-Trøndelag
158 16 15 12 (3–20) 11 (3–18) 12 7 Oppland
162 19 15 24 (8–40) 21 (8–27) 16 12 Oppland
164 14 15 34 (20–51) 26 (20–30) 19 13 Oppland
168 23 15 11 (3–16) 12 (4–19) 6 8 Østfold
174 21 15 19 (4–34) 13 (5–19) 11 8 Østfold
176 20 15 23 (5–45) 19 (4–27) 12 12 Oslo
210 11 15 26 (7–44) 18 (8–23) 18 10 Nord-Trøndelag
214 18 17 18 (6–29) 15 (6–23) 9 9 Sør-Trøndelag
219 15 16 21 (5–34) 19 (4–24) 14 12 Hedmark
223 16 15 20 (7–39) 17 (6–24) 12 9 Hedmark
228 21 15 13 (10–18) 13 (11–17) 8 7 Oslo
237 15 2 45 (42–48) 27 (26–28) 21 10 Sør-Trøndelag
239 18 15 24 (10–48) 20 (10–26) 15 10 Sør-Trøndelag
252 16 5 41 (31–53) 23 (18–25) 22 5 Oppland
253 14 5 18 (11–28) 13 (7–21) 12 5 Oppland
254 15 5 20 (11–42) 13 (8–22) 13 8 Oppland
262 21 15 19 (11–35) 18 (11–26) 10 10 Akershus
348 15 1 12 (12–12) 13 (13–13) 25 3 Aust-Agder
377 13 11 18 (12–25) 17 (11–22) 15 11 Buskerud
402 21 15 22 (10–36) 20 (11–26) 12 10 Vestfold
406 22 15 18 (11–23) 20 (13–24) 10 14 Vestfold
452 13 15 12 (41–6) 10 (3–14) 9 4 Nord-Trøndelag
459 25 15 25 (12–39) 23 (16–27) 10 14 Telemark
512 15 5 16 (12–15) 14 (12–15) 11 9 Oslo
514 11 7 10 (4–14) 10 (4–15) 8 4 Hedmark
519 20 4 12 (10–15) 13 (9–16) 9 7 Hedmark
540 19 4 11 (10–14) 9 (8–11) 7 2 Østfold
593 15 15 16 (8–23) 14 (9–20) 9 5 Troms
611 16 15 11 (3–21) 10 (3–16) 6 4 Nordland
D = diameter at breast height; H = tree height from ground to top
Table 2. Parameters of the fixed-effects Kozak 97 model for stem taper over bark.
Parameter Estimate t value
a0 0.8795844 142.01
a1 0.9457765 244.79
a2 0.1061692 20.06
b1 0.6624965 102.80
b2 0.3781478 50.36
b3 –0.4418430 –21.04
b4 –0.5154988 –37.38
b5 0.0197963 41.52
1

Fig. 1. Stem profiles predicted for a small and a large tree using the three tested taper functions. D is diameter at breast height; H is tree height from ground to top.

Table 3. Parameters of the site index-dependent fixed-effects Kozak 97 model for stem taper over bark.
Parameter Constant Slope coefficient
Estimate t value Estimate t value
a0 1.0233871 30.14 –0.0074925 –4.10
a1 1.1012260 60.36 –0.0084932 –8.80
a2 –0.1160389 –4.30 0.0119276 0.00
b1 0.9717245 35.10 –0.0161964 –11.07
b2 0.0787199 2.44 0.0153552 8.91
b3 –1.4247046 –15.34 0.0488699 10.16
b4 –0.7021483 –11.21 0.0105041 3.29
b5 0.0486118 22.22 –0.0014326 –13.30
Table 4. Parameters of the age-dependent fixed-effects Kozak 97 model for stem taper over bark.
Parameter Constant Slope coefficient
Estimate t value Estimate t value
a0 0.8294 44.76 0.0006427 3.06
a1 0.8600 69.38 0.0008741 7.21
a2 0.2179 13.39 –0.001193 –7.13
b1 0.5449 24.13 0.001253 5.84
b2 0.5581 21.74 –0.001806 –7.32
b3 –0.07002 –0.93 –0.003486 –4.68
b4 –0.4296 –9.07 –0.001174 –2.51
b5 9.725e-03 4.87 8.524e-05 4.57
2

Fig. 2. Effect of site index (SI) on stem taper in a small and a large tree. The continuous line shows the stem taper according to the baseline model (Table 2). D is diameter at breast height; H is tree height from ground to top.

Table 5. Parameters of the mixed-effects Kozak 97 model for stem taper over bark.
Parameter Fixed effect Standard deviation of random effect
Estimate t value
a0 0.9574392 115.43 -
a1 0.9874995 192.41 0.006438417
a2 0.0313538 4.77 -
b1 0.6955883 53.39 0.087608971
b2 0.2987029 49.33 0.079936209
b3 –0.7959772 –68.74 -
b4 –0.3967074 –11.80 -
b5 0.0334238 49.34 0.009906016
Table 6. Correlations of the random parameters of the mixed-effects model.
α1i β1i β2i
β1i –0.014
β2i 0.172 –0.423
β5i 0.180 –0.547 0.187
3

Fig. 3. Residuals of different models as a function of relative height (h/H). View larger in new window/tab.

4

Fig. 4. Bias (predicted diameter – measured diameter) calculated for different relative heights with the fixed-effects model (Fix), fixed part of the mixed-effects model (Mix) and site index-dependent fixed-effects model (Site).

5

Fig. 5. Comparison of the fixed-effects model (Fixed) and fixed part of the mixed-effects model (Mixed). D is diameter at breast height; H is tree height from ground to top.

6

Fig. 6. Stem profiles predicted with the fixed-effects model of this study (new model) and with the model of Gjølberg 1978 (old model). D is diameter at breast height; H is tree height from ground to top.

Table 7. Parameters of the fixed-effects bark model.
Parameter Estimate t value
a0 0.248 31.9
a1 1.060 55.5
a2 –0.529 –21.1
b1 0.379 15.4
b2 0.166 7.3
b3 0.275 2.9
b4 –0.419 –7.9
b5 0.025 9.7
7

Fig. 7. Under bark diameter can be calculated by predicting bark thickness by the bark model (Table 7) and subtracting it from over bark diameter (Table 2). D is diameter at breast height; H is tree height from ground to top.