Table 2 Abbreviations, descriptions and formulas of available variables for individual-tree diameter growth models

Tree size

\({DBH}_{1}\)(\(\mathrm{cm}\))

Initial tree diameter in the period

\({d}_{1i}\)

\({d}_{2i}\) is the ith final tree diameter in the period, \({d}_{1i}\) is the ith initial tree diameter in the period, and \(\mathrm{ln}\) is the natural logarithm.

\({DBH}_{2}\)(\(\mathrm{cm}\))

Final tree diameter in the period

\({d}_{2i}\)

\({DBH}_{N}\)(\(\mathrm{cm}\))

Reciprocal of initial tree diameter

\(1/{\mathrm{d}}_{1i}\)

\({DBH}_{L}\)(cm)

Initial tree logarithmic diameter

\(\mathrm{ln}({\mathrm{d}}_{1i})\)

Stand attributes

\({D}_{g}\)(\(\mathrm{cm}\))

Quadratic mean diameter

\(\sqrt{(\sum {d}_{1i}^{2})/{n}_{j}}\)

\({d}_{1il}\) is the ith initial tree diameter of larch in the period, \({d}_{1ib}\) is the ith initial tree diameter of birch in the period, \({n}_{j}\) is the number of measured trees, \({n}_{jl}\) is the number of measured trees of larch, \({n}_{jb}\) is the number of measured trees of birch, \({h}_{1i}\) is the initial height of tree \(i\) in the period, \({n}_{k}\) is the number of measured trees in plot \(k\) (3–5 trees selected for measurement according to mean diameter), \(\overline{D }\) is the arithmetic mean diameter of trees, and \(S\) is the plot area.

\({D}_{gl}\)(\(\mathrm{cm}\))

Quadratic mean diameter of larch

\(\sqrt{(\sum {d}_{1il}^{2})/{n}_{jl}}\)

\({D}_{gb}\)(\(\mathrm{cm}\))

Quadratic mean diameter of birch

\(\sqrt{(\sum {d}_{1ib}^{2})/{n}_{jb}}\)

\({H}_{p}\)(\(\mathrm{m}\))

Average height of the dominant species

\(\sum {h}_{1i}/{n}_{k}\)

\({A}_{ge}\)

Age of forest stand

-

\(G\)(\({\mathrm{m}}^{2}/\mathrm{ha}\))

Total species stand basal area

\(\frac{10000}{S}\left({\sum }_{i=1}^{{n}_{j}}\frac{\pi }{4}{d}_{1i}^{2}\right)\)

\(\mathrm{Ln}G\)(\({\mathrm{m}}^{2}/\mathrm{ha}\))

Logarithm of \(G\)

\(\mathrm{ln}(G)\)

\({\mathrm{G}}_{l}\)(\({\mathrm{m}}^{2}/\mathrm{ha}\))

Larch species stand basal area

\(\frac{10000}{S}\left({\sum }_{i=1}^{{n}_{j}}\frac{\pi }{4}{d}_{l1i}^{2}\right)\)

\({\mathrm{G}}_{b}\)(\({\mathrm{m}}^{2}/\mathrm{ha}\))

Birch species stand basal area

\(\frac{10000}{S}\left({\sum }_{i=1}^{{n}_{j}}\frac{\pi }{4}{d}_{b1i}^{2}\right)\)

\(N\)(trees/ha)

Number of trees per hectare

\(\frac{10000}{S}{n}_{j}\)

\({N}_{l}\)(trees/ha)

Number of trees of larch per hectare

\(\frac{10000}{S}{n}_{jl}\)

\({N}_{b}\)(trees/ha)

Number of trees of birch per hectare

\(\frac{10000}{S}{n}_{jb}\)

Competition effects

\(BAL\)(\({\mathrm{m}}^{2}/\mathrm{ha}\))

Basal area of larger trees for all species

\(\frac{10000}{S}\left({\sum }_{i=1}^{n}\frac{\pi }{4}{D}_{1q}^{2}\right)\)

\({D}_{1q}\) is the initial diameter of a tree that is larger than the target tree in the period, \({D}_{{1q}_{1}}\) is the initial diameter of a tree that is larger than the intraspecific target tree in the period, \({D}_{{1q}_{2}}\) is the initial diameter of a tree that is larger than the interspecific target tree in the period, and \(n\) is the number of trees.

\({BAL}_{1}\)(\({\mathrm{m}}^{2}/\mathrm{ha}\))

Intraspecific basal area of larger trees

\(\frac{10000}{S}\left({\sum }_{i=1}^{n}\frac{\pi }{4}{D}_{1{q}_{1}}^{2}\right)\)

\({BAL}_{2}\)(\({\mathrm{m}}^{2}/\mathrm{ha}\))

Interspecific basal area of larger trees

\(\frac{10000}{S}\left({\sum }_{i=1}^{n}\frac{\pi }{4}{D}_{1{q}_{2}}^{2}\right)\)

\(BALD\)

Ratio of \(BAL\) and \({d}_{1}\)

\(BAL/\mathrm{ln}({d}_{1}+1)\)

\({BALD}_{1}\)

Ratio of \({BAL}_{1}\) and \({d}_{1}\)

\({BAL}_{1}/\mathrm{ln}({d}_{1}+1)\)

\({BALD}_{2}\)

Ratio of \({BAL}_{2}\) and \({d}_{1}\)

\({BAL}_{2}/\mathrm{ln}({d}_{1}+1)\)

Diversity index

\(SWI\)

Shannon–Wiener index

\(-{\sum }_{k=1}^{m}\frac{{n}_{1jk}}{{n}_{1j}}\mathrm{ln}\left(\frac{{n}_{1jk}}{{n}_{1j}}\right)\)

\(m\) is the number of species, \({n}_{1jk}\) is the initial number of trees for species \(k\) in plot \(j\) in the period, and \({n}_{1j}\) is the total initial number of trees within plot \(j\) in the period.

\(SPI\)

Simpson’s index

\({\sum }_{k=1}^{m}\frac{{n}_{1jk}({n}_{1jk}-1)}{{n}_{1j}({n}_{1j}-1)}\)

\(TAI\)

Total species abundance

\({\sum }_{k=1}^{m}\frac{{n}_{1jk}}{{n}_{1j}}\)

\({\mathrm{G}}_{LR}\)

Larch basal area proportion (%)

\(\frac{{G}_{l}}{G}\)

\({\mathrm{G}}_{BR}\)

Birch basal area proportion (%)

\(\frac{{G}_{b}}{G}\)

Topographic conditions

\(ALT (\mathrm{m})\)

Elevation

-

\(ASP (^\circ )\)

Slope aspect

-

\(SL (^\circ )\)

Slope gradient

-

\({ALTL}_{CA}\)

Logarithmic elevation times cosine of aspect

\(\mathrm{ln}\left(ALT\right)*\mathrm{cos}(ASP)\)

\({ALTL}_{TS}\)

Logarithmic elevation times tangent of slope

\(\mathrm{ln}\left(ALT\right)*\mathrm{tan}(SL)\)