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

Kaev, Taavi; Padari, Allar; Tarmu, Toomas; Kaimre, Paavo; Kiviste, Andres

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

Forestry Studies

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

By:

Taavi Kaev, Allar Padari, Toomas Tarmu, Paavo Kaimre and Andres Kiviste

Open Access

|Oct 2025

Figures & Tables

Locations of ENFRP permanent sample plots.

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.

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 +dc(a + bdc)	dc(h − 1.3)=a + bdc	(Hoßfeld, 1823; Kiviste et al., 2002)	a<=0 \| b<=0	1.3	1.3 +1b	(a(c − 1)b(c + 1))1c
Fla	h=1.3 +dc(a + bdc)	1(h − 1.3)=b + adc	(Hoßfeld, 1823; Kiviste et al., 2002)	a<=0 \| b<=0	1.3	1.3 +1b	(a(c − 1)b(c + 1))1c
F2	h=1.3+(d(a+bd))c	d(h−1.3)1c=a+bd	(Näslund, 1936)	a<=0 \| b<=0	1.3	1.3+1bc	a(c−1)2b
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)=lna+bd	(Curtis, 1967)	a<=0 \| b>=0	1.3	1.3 + a	−b2
F5	h=1.3+a(dd+c)b	ln(h−1.3)=lna+bln(dd+c)	(Curtis, 1967)	a<=0 \| b<=0	1.3	1.3 + a	c(b−1)2
F5a	h=1.3+ad(1+d)b	ln(d(h−1.3))=a+bln(1+d)	(Mehtätalo et al., 2015)	a<=0 \| b<=0	1.3	–	–
F6	h=1.3+ea+b(d+1)	ln(h−1.3)=a+b(d+1)	(Wykoff et al., 1982)	b>=0	1.3 + e ^a+b	1.3+ e^a	−b2−1
F7	h= 1.3 + ade^−bd	ln((h−1.3)d)=lna−bd	(Lebedev, 2020)	a<=0 \| b<=0 \| d>l/b	1.3	1.3	2b
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	lncb
F10	h = a + b log d	-	(Curtis, 1967)	b<=0	– ∞	–	–
F11	h=a+b1d2	–	(Curtis, 1967)	a<=0 \| b>=0	–∞	a	–
F12	h = a + bd	–	Line	b<=0	α	–	–
F13	h=a+bd	–	Hyperbole	a<=0 \| b>=0	–∞	a	–
F14	h=a+bd	–	Square root	a<=0\|b<=0	0	–	–

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

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

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.

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

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Language: English

Page range: 22 - 36

Published on: Oct 30, 2025

Published by: Estonian University of Life Sciences

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

heigh-diameter curve,

ENFRP,

prediction accuracy,

nonlinear regression,

Related subjects:

Ecology,

© 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.

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

Figures & Tables

Figure 1.

Figure 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)_

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)_

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_

The meanings of the column headings of Table 3_

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