Silva Fennica

Fig. 1. Map of northern China indicating the research area of Zhongtiaoshan forest region with a grey large dot (d-maps 2020, modified). East Asia free map, free blank map, free outline map, free base map hydrography, states, main cities, names. Gap https://d-maps.com/carte.php?num_car=28780&lang=en. Accessed 19 January 2021.

Table 1. Tree species-specific information of sample trees of the investigated tree species Pinus tabuliformis, Pinus armandii, Larix principis-rupprechtii, Quercus liaotungensis, Quercus variabilis, Quercus aliena and Betula platyphylla
tree species	sample trees (n)	elevation (m a.s.l.)	tree age (y)		tree height (m)
tree species	sample trees (n)	elevation (m a.s.l.)	mean	sd	mean	sd
P. tabuliformis	27	1187–1677	69	17	14.55	2.40
P. armandii	27	1338–1897	57	12	14.68	2.47
L. principis-rupprechtii	27	1191–1916	34	7	16.86	2.62
Q. liaotungensis	27	1283–1956	57	30	15.88	1.84
Q. variabilis	26	750–1223	60	14	18.20	3.41
Q. aliena	27	932–1350	79	16	17.45	1.76
B. platyphylla	27	1519–1927	38	18	16.18	1.68

Fig. 2. Observed tree heights for all investigated tree species Pinus tabuliformis, Pinus armandii, Larix principis-rupprechtii, Quercus liaotungensis, Quercus variabilis, Quercus aliena and Betula platyphylla from stem analysis. Height of annual shoots are estimated based on the method from Carmean (1972).

Table 2. Model formulations of the two base functions and the five resulting dynamic equations used to fit tree species specific families of height growth curves.
base function	site parameters	solution for X with initial values (y₀, t₀)	dynamic equation	ID
				L1
				L2
				L3
				R1
				R2

Table 3. Akaike information criterion (AIC), root mean square error (RMSE) and bias (ē) for all fitted height growth models based on the five dynamic equations and fitted with both approaches the nonlinear mixed-effects model approach (NLME) and the nonlinear model approach with dummy variables (NLFE) to all investigated tree species Pinus tabuliformis, Pinus armandii, Larix principis-rupprechtii, Quercus liaotungensis, Quercus variabilis, Quercus aliena and Betula platyphylla
tree species	approach	model	AIC	RMSE	ē	approach	model	AIC	RMSE	ē
P. tabuliformis	NLME	L1	–4596.3	34.63	28.79	NLFE	L1	–4465.8	34.61	28.7
		L2	–4541	34.59	28.75		L2	–4446.7	34.6	28.68
		L3	–4337.3	34.4	28.63		L3	–4305.2	34.39	28.55
		R1	–4650.7	34.5	28.63		R1	–4716	34.48	28.56
		R2	–4574.5	34.44	28.58		R2	–4657.8	34.46	28.53
P. armandii	NLME	L1	–2744.4	26.12	21.68	NLFE	L1	–2636.1	26.19	21.69
		L2	–2703.5	26.11	21.67		L2	–2610.7	26.16	21.65
		L3	–2583.1	26.05	21.64		L3	–2525	26.07	21.58
		R1	–2794.8	25.96	21.49		R1	–2818.2	25.99	21.44
		R2	–2739.2	25.94	21.48		R2	–2774	25.97	21.42
L. principis-ruprpechtii	NLME	L1	–380.1	10.68	8.65	NLFE	L1	–332.8	10.76	8.67
		L2	–378.9	10.67	8.64		L2	–333.4	10.78	8.68
		L3	–350.1	10.58	8.58		L3	–324.3	10.67	8.61
		R1	–408.1	10.45	8.37		R1	–475.7	10.5	8.38
		R2	–402.9	10.44	8.36		R2	–437.2	10.47	8.3
Q. liaotungensis	NLME	L1	–2104.9	43.17	33.75	NLFE	L1	–2111.3	43.27	33.68
		L2	–2110.8	43.18	33.73		L2	–2118.6	43.29	33.68
		L3	–2116.8	43.09	33.63		L3	–2169.9	43.28	33.64
		R1	–2210	42.82	33.41		R1	–2232.1	42.68	33.21
		R2	–2200.2	42.86	33.41		R2	–2224.9	42.8	33.28
Q. variabilis	NLME	L1	–1322.1	26.21	21.5	NLFE	L1	–1336.5	26.32	21.56
		L2	–1340.8	26.2	21.46		L2	–1352.2	26.34	21.54
		L3	–1345.6	26.1	21.35		L3	–1370.9	26.24	21.41
		R1	–1415	25.91	21.17		R1	–1492.5	25.94	21.12
		R2	–1439.4	25.91	21.14		R2	–1514	25.99	21.12
Q. aliena	NLME	L1	–2469.6	30.49	25.3	NLFE	L1	–2536.3	30.46	25.2
		L2	–2464.5	30.48	25.29		L2	–2536.2	30.48	25.2
		L3	–2417.8	30.37	25.19		L3	–2501.7	30.38	25.13
		R1	–2555.2	30.28	25.09		R1	–2656.4	30.22	24.97
		R2	–2538	30.27	25.08		R2	–2639.7	30.27	25
B. platyphylla	NLME	L1	–207.1	22.66	16.54	NLFE	L1	–238.3	22.77	16.56
		L2	–187.6	22.64	16.51		L2	–231.7	22.78	16.55
		L3	–118.3	22.49	16.35		L3	–188.6	22.64	16.38
		R1	–224.6	22.53	16.34		R1	–286.4	22.68	16.4
		R2	–199.4	22.51	16.3		R2	–271.2	22.72	16.39

Fig. 3. Plots of the fitted nonlinear model with dummy variables (NLFE) R2 to height growth data from Quercus variabilis. a): family of polymorphic height growth curves with varying asymptotes; numbers after each height growth curve indicate corresponding modelled tree height at base age 40, grey lines in the background indicate observed tree height values. b): residual plots of standardized residuals vs. predicted tree heights. c): predicted tree heights vs. observed tree heights.

Fig. 4. Tree species Pinus tabuliformis, Pinus armandii, Quercus liaotungensis, Quercus aliena and Betula platyphylla are organized in rows, plot types of the fitted nonlinear models with dummy variables (NLFE) R1 to height growth data are organized in columns. First column: families of polymorphic height growth curves with varying asymptotes; numbers after each height growth curve indicate corresponding modelled tree height at base age 40, grey lines in the background indicate observed tree height values. Second column: residual plots of standardized residuals vs. predicted tree heights. Third column: predicted tree heights vs. observed tree heights.

Fig. 5. Plot types of the fitted nonlinear models with dummy variables (NLFE) and nonlinear mixed-effects models (NLME) using the dynamic equations R1, R2 and L1 to height growth data of Larix principis-rupprechtii are organized in columns. First column: families of polymorphic height growth curves with varying asymptotes; numbers after each height growth curve indicate corresponding modelled tree height at base age 40, grey lines in the background indicate observed tree height values. Second column: residual plots of standardized residuals vs. predicted tree heights. Third column: predicted tree heights vs. observed tree heights.

Table 4. Estimated parameter values for the best fitting height growth model of each investigated tree species Pinus tabuliformis, Pinus armandii, Larix principis-rupprechtii, Quercus liaotungensis, Quercus variabilis, Quercus aliena and Betula platyphylla with standard errors, t-statistics and p-values.
tree species	model	parameter	estimate	standard error	t-value	p-value
P. tabuliformis	R1 NLFE	X	2.6284	0.055	47.74	<0.0001
		k	0.0175	0.0007	26.48	<0.0001
		α	3.5901	0.0334	107.38	<0.0001
P. armandii	R1 NLFE	X	2.8004	0.0566	49.43	<0.0001
		k	0.0249	0.0011	23.19	<0.0001
		α	4.2164	0.061	69.1	<0.0001
L. principis-rupprechtii	L1 NLME	X	4.1185	0.0835	49.34	<0.0001
		α	32.7942	0.5528	59.33	<0.0001
		m	0.5198	0.0212	24.52	<0.0001
Q. liaotungensis	R1 NLFE	X	3.1125	0.0713	43.65	<0.0001
		k	0.0167	0.0011	15.02	<0.0001
		α	3.872	0.0675	57.33	<0.0001
Q. variabilis	R2 NLFE	X	3.0037	0.0676	44.4	<0.0001
		k	0.0318	0.0014	23.17	<0.0001
		α	1.1591	0.0335	34.61	<0.0001
Q. aliena	R1 NLFE	X	2.9198	0.0647	45.14	<0.0001
		k	0.0182	0.0011	16.68	<0.0001
		α	3.419	0.0459	74.54	<0.0001
B. platyphylla	R1 NLFE	X	2.6749	0.1198	22.32	<0.0001
		k	0.0326	0.0023	13.88	<0.0001
		α	3.9128	0.0843	46.42	<0.0001