Evaluation of height-diameter equations for predicting dominant height using data from Estonian forest research plots

Distribution of cohorts by canopy layer and tree species (the number of cohorts that include at least one dominant tree height-diameter measurement pair is presented in brackets).

	Upper canopy layer	Secondary canopy layer	Undergrowth	Advanced regeneration
Pinus sylvestris L.	2077 (1968)	10	0	0
Picea abies (L.) H. Karst.	1039 (940)	697	0	47
Betula spp.	993 (706)	58	0	1
Populus tremula L.	157(144)	2	0	0
Alnus glutinosa (L.) Gaertn.	81 (49)	0	0	0
Alnus incana (L.) Moench	60(15)	1	0	0
Salix spp.	22 (8)	0	0	0
Fraxinus excelsior L.	5(5)	5	0	0
Tilia cordata Mill.	2	1	0	0
Acer platanoides L.	2	8	0	1
Quercus robur L.	1(1)	0	0	0
Sorbus aucuparia L.	0	0	1	0
Corylus avellana L.	0	0	7	0

Table 2.

The standardized height-diameter equations evaluated in the study (h – tree height (m); d – tree diameter at breast height (cm); a, b – estimated parameters; c – parameter fixed as constant).

No	Equation	Linear form	References	Decreasing region	Height at zero diameter	Height asymptote	Diameter at the inflection point
Fl	$h = 1.3 + \frac{d^{c}}{(a + b d^{c})}$	$\frac{d^{c}}{(h - 1.3)} = a + b d^{c}$	(Hoßfeld, 1823; Kiviste et al., 2002)	a<=0 \| b<=0	1.3	$1.3 + \frac{1}{b}$	${(\frac{a (c - 1)}{b (c + 1)})}^{\frac{1}{c}}$
Fla	$h = 1.3 + \frac{d^{c}}{(a + b d^{c})}$	$\frac{1}{(h - 1.3)} = b + \frac{a}{d^{c}}$	(Hoßfeld, 1823; Kiviste et al., 2002)	a<=0 \| b<=0	1.3	$1.3 + \frac{1}{b}$	${(\frac{a (c - 1)}{b (c + 1)})}^{\frac{1}{c}}$
F2	$h = 1.3 + {(\frac{d}{(a + b d)})}^{c}$	$\frac{d}{{(h - 1.3)}^{\frac{1}{c}}} = a + b d$	(Näslund, 1936)	a<=0 \| b<=0	1.3	$1.3 + \frac{1}{b^{c}}$	$\frac{a (c - 1)}{2 b}$
F3	h =1.3 + ad^b	ln(h – 1.3) = ln a + b ln d	(Curtis, 1967)	a<=0 \| b<=0 \|b>l	1.3	–	–
F4	h = 1.3 + ae^b/d	$\ln (h - 1.3) = \ln a + \frac{b}{d}$	(Curtis, 1967)	a<=0 \| b>=0	1.3	1.3 + a	$\frac{- b}{2}$
F5	$h = 1.3 + a {(\frac{d}{d + c})}^{b}$	$\ln (h - 1.3) = \ln a + b \ln (\frac{d}{d + c})$	(Curtis, 1967)	a<=0 \| b<=0	1.3	1.3 + a	$\frac{c (b - 1)}{2}$
F5a	$h = 1.3 + \frac{a d}{{(1 + d)}^{b}}$	$\ln (\frac{d}{(h - 1.3)}) = a + b \ln (1 + d)$	(Mehtätalo et al., 2015)	a<=0 \| b<=0	1.3	–	–
F6	$h = 1.3 + e^{a + \frac{b}{(d + 1)}}$	$\ln (h - 1.3) = a + \frac{b}{(d + 1)}$	(Wykoff et al., 1982)	b>=0	1.3 + e ^a+b	1.3+ e^a	$\frac{- b}{2} - 1$
F7	h= 1.3 + ade^−bd	$\ln (\frac{(h - 1.3)}{d}) = \ln a - b d$	(Lebedev, 2020)	a<=0 \| b<=0 \| d>l/b	1.3	1.3	$\frac{2}{b}$
F8	h= 1.3 + a(ln(l + d))^d	ln(h – 1.3) = ln a + b ln(ln(l + d))	(Lebedev, 2020)	a<=0 \| b<=0	1.3	–	–
F9	h =1.3 + a(l – e^−bd)^c	–	(Chapman, 1961; Meyer, 1940; Nigul et al., 2021; Richards, 1959)	a<=0 \| b<=0	1.3	1.3 + a	$\frac{\ln c}{b}$
F10	h = a + b log d	-	(Curtis, 1967)	b<=0	– ∞	–	–
F11	$h = a + b \frac{1}{d^{2}}$	–	(Curtis, 1967)	a<=0 \| b>=0	–∞	a	–
F12	h = a + bd	–	Line	b<=0	α	–	–
F13	$h = a + \frac{b}{d}$	–	Hyperbole	a<=0 \| b>=0	–∞	a	–
F14	$h = a + b \sqrt{d}$	–	Square root	a<=0\|b<=0	0	–	–

Table 3.

Goodness of fit statistics of the two-parameter height-diameter curves used in the study. The meanings of the column headings are presented in Table 4.

No	Noc	c	Nfail	Ndecr	Nlow	Nhd	RMSE	MAD	ME	ME5%	MEdom	MEDdom	a	b
Fl	1	1.3	0	68	0	127034	1.406	1.202	-0.0043	0.0034	-0.0805	-0.0796	0.7829	0.0365
Fl	2	1.5	1	33	0	127034	1.401	1.198	-0.0013	0.0070	-0.0312	-0.0261	1.1954	0.0393
Fl	3	1.5117	1	33	0	127023	1.401	1.198	-0.0011	0.0072	-0.0282	-0.0231	1.2262	0.0395
Fl	4	1.58	1	30	0	127034	1.401	1.198	-0.0001	0.0085	-0.0108	-0.0033	1.4226	0.0404
Fl	5	1.5896	0	29	0	127034	1.401	1.197	0.0000	0.0086	-0.0084	-0.0013	1.4519	0.0405
Fl	6	1.59	0	29	0	127034	1.400	1.197	0.0000	0.0086	-0.0083	-0.0013	1.4532	0.0405
Fl	7	1.6	0	29	0	127034	1.400	1.197	0.0002	0.0088	-0.0057	0.0008	1.4855	0.0407
Fl	8	1.6118	0	29	0	127034	1.400	1.197	0.0004	0.0090	-0.0027	0.0035	1.5250	0.0408
Fl	9	1.6655	0	28	0	127034	1.400	1.197	0.0012	0.0100	0.0112	0.0187	1.7148	0.0414
Fl	10	1.7	0	28	0	127034	1.400	1.197	0.0017	0.0107	0.0202	0.0277	1.8505	0.0417
Fl	11	1.7128	0	28	0	127034	1.400	1.197	0.0019	0.0109	0.0235	0.0318	1.9038	0.0419
Fl	12	2	0	22	0	127034	1.405	1.200	0.0064	0.0161	0.0998	0.1067	3.5648	0.0444
F1a	13	2	0	22	0	127034	1.405	1.200	0.0064	0.0161	0.0998	0.1067	3.5648	0.0444
Fl	14	3	1	27	6	127024	1.460	1.260	0.0220	0.0313	0.3722	0.3642	35.725	0.0496
F2	15	1	1	304	0	127025	1.419	1.221	-0.0084	-0.0013	-0.1499	-0.1605	0.4339	0.0308
F2	16	2	0	22	0	127034	1.405	1.202	-0.0019	0.0062	-0.1022	-0.1036	0.8874	0.1844
F2	17	3	0	21	0	127034	1.404	1.201	0.0000	0.0086	-0.0838	-0.0808	0.9421	0.3274
F2	18	4	0	21	0	127034	1.403	1.199	0.0009	0.0097	-0.0742	-0.0685	0.8905	0.4343
F2	19	5	1	21	0	127025	1.403	1.200	0.0015	0.0104	-0.0682	-0.0610	0.8180	0.5140
F2	20	10	4	21	0	126925	1.404	1.203	0.0025	0.0115	-0.0560	-0.0458	0.5417	0.7183
F2	21	20	7	19	0	126909	1.405	1.205	0.0029	0.0121	-0.0498	-0.0386	0.3109	0.8479
F2	22	200	91	14	0	125759	1.404	1.204	0.0034	0.0124	-0.0450	-0.0332	0.0354	0.9837
F2	23	1000	1030	2	0	108218	1.389	1.192	0.0040	0.0132	-0.0458	-0.0333	0.0073	0.9967
F2	24	2000	2055	0	0	85463	1.372	1.178	0.0049	0.0140	-0.0517	-0.0208	0.0036	0.9984
F3	25	–	4	310	0	126977	1.448	1.251	-0.0062	0.0022	-0.2843	-0.2888	4.8087	0.4344
F4	26	–	8	21	0	126839	1.406	1.206	0.0034	0.0126	-0.0434	-0.0319	26.941	-7.1578
F5	27	1	3	21	0	126849	1.405	1.204	0.0022	0.0113	-0.0595	-0.0519	27.433	7.6564
F5	28	1	1	21	57	127025	1.407	1.206	0.0029	0.0119	-0.0477	-0.0362	28.308	7.0243
F3	29	–	2	125	8	127007	1.443	1.243	-0.0044	0.0040	-0.2785	-0.2835	5.6670	0.4007
F4	30	–	6	21	100	126864	1.409	1.212	0.0038	0.0130	-0.0311	-0.0194	27.867	-6.5447
F6	31	1	0	21	0	127034	1.404	1.200	0.0012	0.0100	-0.0752	-0.0727	3.3291	-8.1500
F5	32	1.1	3	21	0	126849	1.405	1.204	0.0021	0.0112	-0.0609	-0.0540	27.468	7.0015
F5	33	1.3	2	21	0	126986	1.404	1.201	0.0020	0.0110	-0.0638	-0.0574	27.536	5.9887
F5	34	8	0	20	0	127034	1.407	1.204	-0.0012	0.0069	-0.1285	-0.1388	30.173	1.3628
F5	35	0.5	6	21	0	126758	1.406	1.206	0.0027	0.0119	-0.0518	-0.0411	27.192	14.808
Fl	36	1.59	1	21	0	127034	1.400	1.197	0.0000	0.0086	-0.0083	-0.0013	17.143	0.2572
Fl	37	1.6	0	22	0	127034	1.400	1.197	0.0002	0.0088	-0.0057	0.0008	17.145	0.2553
F5a	38	1	12	285	0	126822	1.444	1.247	-0.0071	0.0011	-0.2627	-0.2675	5.5728	0.6048
F7	39	–	2	282	0	127013	1.414	1.213	-0.0045	0.0024	0.0617	0.0654	1.7051	0.0293
F8	40	–	3	20	0	126997	1.426	1.230	-0.0040	0.0040	-0.2284	-0.2337	4.0636	1.3151
F9	41	1	327	187	0	121663	1.417	1.220	-0.0075	0.0004	-0.0475	-0.0516	23.967	0.0839
F9	42	0.80312	856	23	0	109582	1.410	1.216	-0.0062	0.0033	-0.0946	-0.1033	24.840	0.0669
F9	42	1.18444	242	66	0	123709	1.410	1.210	-0.0049	0.0028	-0.0023	-0.0064	23.109	0.0988
F9	44	1.19374	230	65	0	123885	1.409	1.210	-0.0048	0.0029	-0.0001	-0.0039	23.090	0.0993
F9	45	1.19	233	67	0	123846	1.409	1.210	-0.0048	0.0029	-0.0010	-0.0050	23.102	0.0991
F10	46	–	0	3262	55	127034	1.420	1.221	0.0000	0.0080	-0.1941	-0.1922	-1.5402	7.1452
F1l	47	–	0	21	226	127034	1.590	1.340	0.0000	-0.0015	0.4462	0.4068	21.162	-752.43
F12	48	–	0	136	ü	127034	1.493	1.289	0.0000	0.0118	-0.4066	-0.4012	10.427	0.4127
F13	49	–	0	21	172	127034	1.466	1.249	0.0000	0.0046	0.1151	0.1165	25.051	-108.27
F14	50	–	0	1585	11	127034	1.443	1.245	0.0000	0.0090	-0.3161	-0.3112	3.2249	3.4302

Table 4.

The meanings of the column headings of Table 3.

	Meaning
No	Height-diameter equation number (as presented in Table 2)
Noc	Height-diameter curve number
Nfail	Number of cohorts for which the non-linear regression did not converge.
Ndecr	Number of cohorts for which the estimates for parameters a and b resulted in the height-diameter curve decreasing.
Nlow	Number of height-diameter measurement pairs for which the model predicted a height lower than 1.3 m.

The following characteristics were calculated from the trimmed dataset, from which cohorts with extreme distributions were excluded:
Nhd	The number of height-diameter measurement pairs at the individual tree level.
RMSE	The root mean square error of height prediction.
MAD	The median absolute error of height prediction.
ME	The mean error based of height prediction.
ME5%	The trimmed mean error of height prediction error (excluding the largest and smallest 5% of the data).
MEdom	The mean height prediction error based on dominant tree measurement data.
MEDdom	The median value of height prediction error based on dominant tree measurement data.
a	The median values of parameter a, estimated for cohorts with non-linear regression
b	The median values of parameter b, estimated for cohorts with non-linear regression

Median errors of dominant tree height measurement predictions by cohort mean diameter classes for each out of 15 selected height-diameter curves. The numbers of dominant tree height measurements by mean diameter classes are presented at the bottom of the plots. The height-diameter curve numbers (*Noc* in Table 3) in the legend are in ascending order of the absolute value of the median prediction error of dominant height.

References

Authors

Metrics

Articles in this issue

DOI: https://doi.org/10.2478/fsmu-2024-0011 | Journal eISSN: 1736-8723 | Journal ISSN: 1406-9954

Journal RSS Feed

Language: English

Page range: 22 - 36

Published on: Oct 30, 2025

Published by: Estonian University of Life Sciences

In partnership with: Paradigm Publishing Services

Keywords:

heigh-diameter curve,

ENFRP,

prediction accuracy,

nonlinear regression,

linearization

Related subjects:

Life sciences,

Plant science

© 2025 Taavi Kaev, Allar Padari, Toomas Tarmu, Paavo Kaimre, Andres Kiviste, published by Estonian University of Life Sciences
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 81 (2024): Issue 1 (December 2024)

Evaluation of height-diameter equations for predicting dominant height using data from Estonian forest research plots

Figures & Tables

Figure 1.

Table 1.

Table 2.

Table 3.

Table 4.

Figure 2.

Paradigm

My account