Height–diameter relationship of plantation-grown juvenile black locust trees is differentiated according to their growth rate, which is positively affected by spacing

Sample plot	Parental genotype	Geographic region	Plot size (m²)	Growing space (m²)	Age (years)	Number of measured trees	d (cm)	h (m)
PP 1	–	southern	185.00	2.5	6	48	3.6–7.1	4.7–6.3
PP 2	–	southern	173.25	2.5	2	51	3.1–6.0	4.7–6.5
PP 3	–	southern	202.50	2.5	3	50	3.8–4.6	6.4–6.5
PP 4	–	southern	165.00	3.5	4	67	1.5–4.6	2.9–4.7
PP 5	Roszin Varga	northern	304.00	4.5	5	64	2.3–6.7	2.8–7.7
PP 6	Karaisen	northern	306.04	4.5	5	52	2.4–7.6	3.9–9.2
PP 7	Tsarevets	northern	300.00	4.5	5	70	2.2–9.1	4.2–9.2

№	Model		Mixed-effects parameters	df	logLik	Test	Likelihood Ratio	p-value
№	abbreviation	localisation level	Mixed-effects parameters	df	logLik	Test	Likelihood Ratio	p-value
1	GR.PP.01	sample plot within geographic region	b₀, b₁	9	–413.990
2	PP.01	sample plot	b₀, b₁	6	–414.324	1 vs. 2	0.669	0.881
3	PP.0	sample plot	b₀	4	–419.808	2 vs. 3	10.967	0.004
4	GR.PP.0	sample plot within geographic region	b₀	5	–419.407	3 vs. 4	0.800	0.371
5	PP.1	sample plot	b₁	4	–430.327	4 vs. 5	21.839	<0.001
6	GR.PP.1	sample plot within geographic region	b₁	5	–430.130	5 vs. 6	0.393	0.531
7	GR.01	geographic region	b₀, b₁	6	–438.471	6 vs. 7	16.682	<0.001
8	GR.0	geographic region	b₀	4	–440.748	7 vs. 8	4.553	0.103
9	GR.1	geographic region	b₁	4	–445.830
10	simple local	no localisation	–	3	–453.059	9 vs 10	14.457	<0.001

Plot	u	Relative errors					MARE%	Bias
Plot	u	Perc0	Perc25	Perc50	Perc75	Perc100	MARE%	Bias
PP 1	–0.789	–0.719	–0.150	–0.077	–0.003	0.166	11.829	–0.363
PP 2	–0.561	–0.579	–0.133	–0.037	0.016	0.205	11.001	–0.233
PP 3	0.369	–0.100	–0.018	0.017	0.076	0.153	5.423	0.163
PP 4	–0.981	–0.350	–0.182	–0.067	0.063	0.391	14.612	–0.228
PP 5	–0.125	–0.405	–0.116	–0.011	0.086	0.198	10.536	–0.050
PP 6	1.118	–0.390	–0.047	0.086	0.167	0.242	13.004	0.506
PP 7	0.968	–0.673	–0.025	0.068	0.118	0.238	12.661	0.403

Mixed-effects model: h = 1.3 + (u + b₀)e^b₁
ME 0.773	Model parameters			Model variances
		b₀	b₁	$σ_{u}^{2}$ \sigma _u^2		σ²
MARE% 10.322	Estimate	8.302	–2.631	0.639		0.555
Bias 0.010	SE	0.395	0.146	0.363		0.042
Generalised deterministic model: h = 1.3 + (a₀ + a₁Spacing)e^b₁/d
ME 0.724	Variance function		Model parameters
	(θ+d^η)²			a₀	a₁	b₁
MARE% 11.634	Variance function parameters		Estimate	7.322	0.422	–2.918
Bias 0.025	Θ	H	SE	0.417	0.087	0.123
	37.243	1.701

Model		Mean Bias (m)	Aggregated MSE (m)^*	90^th percentile of ARB_i^*
abbreviation	formula	Mean Bias (m)	Aggregated MSE (m)^*	90^th percentile of ARB_i^*
M1	h = 1.3 + a₁d	0.1515	6.9068	0.2184
M2	h = 1.3 + a₁d + a₂d²	–0.0024	3.8496	0.1367
M3	h = a₀d^a₁	–0.0031	4.0965	0.1176
M4	h = 1.3 + exp[a₀ + a₁/(d + 1)]	–0.0034	3.8616	0.1100
M5	h = 1.3 + a₀ exp(a₁/d)	–0.0005	3.7790	0.1077
M6	h = 1.3 + a₀d^a₁	–0.0053	4.1714	0.1191
M7	h = 1.3 + a₀d/(a₁ + d)	0.1221	6.9214	0.2412