Mohsen Pirmoradiyan
← Geospatial & GIS

Forest structure from above: from tree heights and crowns to DBH, basal area and Lorey's height

What the standard forest numbers (DBH, basal area, stems per hectare, DGV, Lorey's height, dominant height) mean, how to compute them right, and how published models guess DBH and biomass from the tree height and crown size you can see from above.

First explored September 2025·9 min read

forestrylidarphotogrammetryallometrydbhbasal-arearemote-sensing

A forester describes a stand with a handful of numbers: how thick the trees are, how many there are per hectare, how much ground their stems cover, and how tall they are. Most of these are defined at breast height, 1.3 m above the ground. And that’s exactly the part of the tree a drone or an airborne laser scanner can’t see. From above, you get tree height and crown size. So here I collected the standard definitions, fixed a few formulas that are easy to get wrong, and looked at published models that try to bridge the gap between “what we can see” and “what foresters need”.

The basic tree measurement: DBH

DBH (diameter at breast height, written dd or D1.3D_{1.3}) is the stem’s diameter 1.3 m above the ground, measured in centimetres with a calliper or a diameter tape. Pretty much everything else in forest measurement is built on it: basal area, volume, biomass and growth.

The other per-tree values are simpler. Height hh is from the ground to the tip. Crown base height (CBH) is the height of the lowest live branch, so crown length is CL=h−CBHCL = h - \text{CBH} and crown ratio is CR=CL/hCR = CL / h, a number between 0 and 1. Crown width is how wide the crown spreads, usually the average of two measurements at right angles.

Stand numbers, done carefully

Basal area gg of one tree is the area of its stem cut across at 1.3 m. With dd in centimetres, the radius in metres is d/200d/200, so

g=π(d200)2=π40 000 d2≈0.00007854 d2[m2].g = \pi \left(\frac{d}{200}\right)^2 = \frac{\pi}{40\,000}\, d^2 \approx 0.00007854\, d^2 \quad [\text{m}^2].

The stand’s basal area GG is the sum over the trees in a plot, divided by the plot area AA in hectares:

G=1A∑i=1ngi[m2/ha].G = \frac{1}{A}\sum_{i=1}^{n} g_i \quad [\text{m}^2/\text{ha}].

It’s easy to forget the division by AA. Without it you get square metres per plot, not per hectare.

Stems per hectare is N=n/AN = n / A on a fixed-area plot. Angle-count (relascope) sampling works differently, and this took me a moment: every counted tree stands for BAF/gi\text{BAF}/g_i stems per hectare, where BAF is the basal area factor of the gauge. So G=BAF×(count)G = \text{BAF} \times (\text{count}) and N=∑iBAF/giN = \sum_i \text{BAF}/g_i. Thin trees count for many stems, thick trees for few.

Mean diameters. Three are in common use, and they’re not the same number:

  • the arithmetic mean dˉ=1n∑di\bar d = \frac{1}{n}\sum d_i;
  • the quadratic mean diameter Dq=1n∑di2D_q = \sqrt{\frac{1}{n}\sum d_i^2}, the diameter of the tree with average basal area;
  • the basal-area-weighted mean diameter, called DGV in Swedish forest data (grundytevägd medeldiameter):
DGV=∑gidi∑gi=∑di3∑di2.\text{DGV} = \frac{\sum g_i d_i}{\sum g_i} = \frac{\sum d_i^3}{\sum d_i^2}.

Each tree’s diameter is weighted by its basal area, so big trees count more. You’ll sometimes see a shortcut written as 0.00007854∑di3/G0.00007854 \sum d_i^3 / G. That’s only right if GG is the plot sum of gig_i, not the per-hectare value. The ratio ∑d3/∑d2\sum d^3 / \sum d^2 is safer.

Heights. The plain mean height hˉ\bar h gets dragged down by small trees stuck under the canopy. Lorey’s height weights each tree by its basal area, just like DGV:

HL=∑gihi∑gi=∑di2hi∑di2.H_L = \frac{\sum g_i h_i}{\sum g_i} = \frac{\sum d_i^2 h_i}{\sum d_i^2}.

Swedish data call it HGV (grundytevägd medelhöjd). It follows stand volume better than hˉ\bar h. Dominant height HdomH_{\text{dom}} is the mean height of the 100 thickest trees per hectare. On a plot of AA hectares that means the 100A100A thickest trees, so 10 trees on a 0.1 ha plot. Dominant height barely changes when a stand is thinned, which is why it’s used for site index: the dominant height a stand reaches at a reference age. It’s often modelled with a Chapman–Richards curve Hdom=a(1−e−bt)cH_{\text{dom}} = a(1 - e^{-bt})^c, where tt is stand age, aa the height it levels off at, bb a rate and cc a shape parameter.

Reineke’s stand density index compares the stem count to that of a stand with a reference mean diameter of 25 cm:

SDI=N(Dq25)1.605.\text{SDI} = N \left(\frac{D_q}{25}\right)^{1.605}.

The exponent is positive. A version with −1.605-1.605 floats around in some notes, but it makes a stand of big trees look less dense, which is the opposite of the whole point.

Here’s all of it on a synthetic plot:

import numpy as np

rng = np.random.default_rng(0)
A = 0.1                                       # plot area, ha
d = rng.gamma(shape=9, scale=2.6, size=60)    # DBH, cm
h = 1.3 + (d / (1.6 + 0.20 * d)) ** 2         # Näslund-type height curve, m

g   = np.pi * (d / 200) ** 2                  # basal area per tree, m²
G   = g.sum() / A                             # m²/ha
N   = len(d) / A                              # stems/ha
Dq  = np.sqrt((d ** 2).mean())
DGV = (g * d).sum() / g.sum()                 # = Σd³ / Σd²
H_L = (g * h).sum() / g.sum()                 # Lorey's height
H_dom = h[np.argsort(d)[-round(100 * A):]].mean()
SDI = N * (Dq / 25) ** 1.605
# N=600, G=27.5, mean d=23.1, Dq=24.2, DGV=27.2, mean h=14.7, H_L=15.9, H_dom=17.7

It’s always dˉ≤Dq≤DGV\bar d \le D_q \le \text{DGV}, and the gap gets bigger the more the diameters vary. The height model above is Näslund’s function h=1.3+d2/(a+bd)2h = 1.3 + d^2/(a + b d)^2, the classic height–diameter curve in Nordic forestry.

Left: histogram of DBH in a synthetic plot with the arithmetic mean, quadratic mean and basal-area-weighted mean marked; DGV is the largest. Right: predicted versus true DBH for a height-only model and a height plus crown area model; the second scatter is tighter around the 1:1 line.

Bridging the gap: guessing DBH from height and crown

From a canopy height model and a crown segmentation you get, per tree, the height HH and the crown area CACA (or radius RR). Allometry is the idea that parts of a tree grow in proportion to each other, roughly following power laws. So a natural model is

DBH=b0 (H−1.3)b1 CAb2,\text{DBH} = b_0\,(H - 1.3)^{b_1}\, CA^{b_2},

which turns into a straight-line model once you take logs. Four published studies show the same lesson from different angles.

Chen et al. (2007), blue oak woodland in California, small-footprint LiDAR. They compared lots of models for single-tree basal area and stem volume. A linear DBH model from maximum height and crown area, DBH=2.29+1.6 H100+0.26 Ca\text{DBH} = 2.29 + 1.6\,H_{100} + 0.26\,C_a (DBH in cm, H100H_{100} the highest return in m, CaC_a the crown area in m²), followed by B=π(DBH/2)2B = \pi(\text{DBH}/2)^2, was among the best when crowns were segmented correctly. Their own new idea was the canopy geometric volume VcV_c, the volume under the crown surface. They modelled basal area and stem volume as simple power functions of VcV_c, and these held up best when the segmentation made mistakes.

Cao & Dean (2013) start from the uniform stress theory. The idea is that a stem tapers so that wind pushing on the crown causes about the same bending stress all along the stem. So the diameter at some height depends on the leaf area above it times the lever arm to the centre of that leaf area. With HMC=H−CL/2H_{MC} = H - CL/2 (height of the crown centre) and the lever arm L=HMC−1.37L = H_{MC} - 1.37 m, they fit D=b1(AL⋅L)b2D = b_1 (A_L \cdot L)^{b_2}, where the leaf area ALA_L is itself estimated from height and crown length. Both inputs come from LiDAR. Models with crown length clearly beat height-only power models, and adding stems per hectare helped all of them.

Gonzalez-Benecke et al. (2022), plantations of Pinus radiata, Eucalyptus globulus and E. nitens, built three levels of models for DBH (cm), stem volume and above-stump biomass: height only; height plus crown area; and height, crown area and stand variables like age and stems per hectare. Height-only models worked, but their errors grew with tree size. Adding crown area gave the biggest single improvement and got rid of that size bias.

Liu et al. (2024) skip DBH and predict above-ground biomass WW directly from height HH and crown radius RR measured from UAV data. The reference biomass came from standard W=a(D2H)bW = a(D^2H)^b equations. They tried nine forms. Models that used both HH and RR beat models with just one of them, and the quadratic W=a(H+R)2+b(H+R)+cW = a(H+R)^2 + b(H+R) + c was the most consistent across forest types (in mixed coniferous stands, R2≈0.93R^2 \approx 0.93).

On synthetic trees where crown size carries real information, the effect is easy to see:

rng = np.random.default_rng(1)
H  = rng.uniform(8, 30, 300)                                   # tree height, m
CA = np.exp(rng.normal(np.log(0.12 * H**1.5), 0.35))           # crown area, m²
DBH = 1.2 * (H - 1.3)**0.75 * CA**0.30 * np.exp(rng.normal(0, 0.08, 300))  # cm

def fit_power(y, *xs):
    """Fit y = b0 * x1^b1 * x2^b2 ... by least squares on logs."""
    X = np.column_stack([np.ones_like(y)] + [np.log(x) for x in xs])
    b, *_ = np.linalg.lstsq(X, np.log(y), rcond=None)
    return b, np.exp(X @ b)

b1, pred1 = fit_power(DBH, H - 1.3)          # height only     -> RMSE 2.9 cm
b2, pred2 = fit_power(DBH, H - 1.3, CA)      # + crown area    -> RMSE 1.8 cm

Two things to watch out for. First, turning a log-scale fit back with exp gives you the median, not the mean, so it comes out a bit low. A correction factor like eσ2/2e^{\sigma^2/2} fixes most of that. Second, every coefficient is specific to a species and region. A model fitted to plantation eucalyptus tells you nothing about Swedish spruce until you refit it on local field plots.

From trees to stand values

The literature has two broad routes. In the individual-tree approach (for example Hyyppä et al., 2001), each detected tree gets a predicted DBH from models like the ones above. The stand numbers then come from summing gig_i, di3d_i^3 or di2hid_i^2 h_i over the trees in an area and dividing by that area. Its weak spots are trees hidden under the canopy, which never get detected, and crowns that get merged or split, which mess up both the tree count and the crown areas (see polygon validity and shape metrics for checking crown shapes). In the area-based approach (Næsset, 2002), stand values like GG, NN or volume are regressed straight on height statistics of the point cloud over field plots, with no tree detection at all. Hidden trees don’t hurt it much, but you don’t get a list of trees. When volume per tree is needed, it’s usually a species function v=a dbhcv = a\, d^b h^c.

Key takeaways

  • Basal area per tree is π(d/200)2\pi (d/200)^2 m² with dd in cm. Divide the plot sum by plot area to get m²/ha.
  • dˉ\bar d, DqD_q and DGV are three different means. DGV and Lorey’s height both weight trees by basal area (∑d3/∑d2\sum d^3/\sum d^2 and ∑d2h/∑d2\sum d^2 h/\sum d^2).
  • Dominant height uses the 100 thickest trees per hectare. Reineke’s SDI uses a positive exponent of 1.605.
  • Height alone predicts DBH and biomass with errors that grow for big trees. Adding crown size (area, radius or length) fixes a lot of that.
  • Allometric coefficients are local: refit them with field data before trusting them.

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