Forestry & Forest Sciences · Mensuration, Inventory & Geospatial Analysis · also Forest Measurements, Forest Mensuration and Biometry, Dendrometry
Measuring single trees, logs and stands: diameter, height, form, volume, biomass and growth, with the instruments and formulas used in the field.
In the exams: IFoS Forestry Paper II Section A (forest mensuration); ICAR AICE-JRF 59 Unit 4; ICAR AIEEA-PG Group 09 (forest mensuration); PRC Forest Resource Management; SAF Certified Forester; state forester and RFO exams
Forest mensuration is the measurement of trees, logs and stands: their diameter, height, shape, volume, biomass, age and growth. The word comes from the Latin mensura, a measure. Every other part of forest management leans on it. A working plan cannot set an allowable cut without an estimate of growing stock and increment; a timber sale cannot be priced without a volume; a carbon project cannot be credited without a biomass estimate; and a silviculturist cannot judge whether a thinning worked without remeasuring the stand. Mensuration turns a forest into numbers that can be compared, audited and planned with.
The subject has three layers. The first is single-tree measurement: diameter at breast height, total and merchantable height, bark thickness, crown size, and the volume of logs and standing trees. The second is the link between easy and hard measurements: form factors, volume tables and equations, and allometric biomass models, which let a forester estimate volume or biomass from diameter and height alone. The third is the stand and its change over time: stems per hectare, basal area, top height, stand volume, angle-count sampling, age, increment and site quality. Forest inventory and sampling design, which decide where and how many plots to measure, are a separate subject in this track (forest-inventory-sampling), as are growth and yield models (growth-yield-modelling); this manual gives the measurement and estimation foundations both rely on.
Mensuration is examined everywhere forestry is examined. In India it is a full unit of the ICAR AICE-JRF Forestry syllabus and of the ICAR AIEEA-PG forestry group, and a named section of Paper II of the Indian Forest Service forestry optional; state forest-service and range-officer examinations set short numerical problems on log volume, basal area and increment. The Philippine Foresters licensure examination places it under Forest Resource Management, Korea's Forestry Engineer qualification under forest management, and professional credentials in North America assume it as routine field competence. The numerical archetypes are few and repeat: girth to diameter, basal area and its per-hectare expansion, the Huber, Smalian and Newton formulas, form factor, height from angles or percent slopes, basal area factor and limiting distance, mean and current annual increment, increment percent, and site index from a guide curve. This manual works each of them and points out the slips that cost marks.
Conventions differ between countries, and this manual says so where they matter: breast height is 1.3 m in most metric countries but 4.5 ft (1.37 m) in the United States and traditionally in India; some countries record girth rather than diameter; top height has more than one definition; and carbon fractions and expansion factors have default values that vary by guideline. In an examination, use the convention the question states.
Diameter is the most measured quantity in forestry because it is easy to measure accurately and closely related to volume, biomass and value. It is measured at a fixed reference height so that measurements are comparable between trees, crews and decades. That height, breast height, was chosen to be convenient for a standing person and to lie above most of the root swelling at the stem base. Diameter at breast height outside bark is written dbh, DBH or d1.3.
The convention is not the same everywhere. Most countries using metric units take breast height as 1.3 m above ground. The United States uses 4.5 ft, which is 1.37 m. India has traditionally used 1.37 m (4 ft 6 in) as well, and Indian practice often records girth at breast height rather than diameter, so many Indian volume tables are entered by girth. A few countries and older tables use 1.4 m or other heights. Whatever the convention, it must be stated on the data sheet, and a volume table must be entered with diameters measured the same way it was built. An examination question that gives a height convention expects you to use it.
Ground level is taken on the uphill side of a tree standing on a slope. A leaning tree is measured at breast height along the stem, not vertically, and the diameter is taken at right angles to the stem axis. A tree that forks below breast height is treated as two (or more) trees, each measured at breast height; a tree that forks above breast height is one tree measured normally. If there is a swelling, branch, wound or burl at breast height, the diameter is taken just above it (or the average of measurements just above and just below, depending on the national rule). Buttressed tropical trees, whose flared bases can rise several metres, are measured a fixed distance above the top of the buttress, and the point of measurement is recorded so the tree can be remeasured at the same place. Painting or marking the point of measurement on permanent plots is standard practice, because remeasurement at a slightly different height can create apparent growth or shrinkage larger than the true increment.
Two instruments dominate. Tree calipers have a fixed arm and a sliding arm on a graduated beam; the beam is held at right angles to the stem and both arms touch the bark. Because few stems are perfectly round, two readings at right angles are usually averaged. Calipers are quick and do not need to go round the tree, but the arms must be parallel and pressed lightly, and the beam must be perpendicular to the stem; a tilted beam or loose arm adds error.
A girth tape is wrapped around the stem at breast height, kept horizontal (perpendicular to the stem axis), and read where it overlaps. A diameter tape is the same tape graduated in units of π, so that each 1 cm of diameter takes 3.14 cm of tape and diameter can be read directly. Diameter is girth divided by π: a girth of 157 cm is a diameter of 50 cm. Quarter girth, girth divided by four, is not a diameter; it belongs to the old Hoppus volume rule discussed later.
Tape and calipers do not give identical results on irregular stems. For an elliptical cross-section, the perimeter is larger than that of a circle of the same area, so diameter from a tape (perimeter divided by π) overestimates the diameter of the circle of equal area, and basal area computed from a tape diameter is slightly too large. Calipers averaged over two perpendicular axes give a smaller bias. On permanent plots tapes are often preferred because they give more repeatable results, and the small bias is the same at each measurement.
Diameter is usually recorded to the nearest millimetre or half-centimetre on research plots and to the nearest centimetre or in 2-cm or 5-cm classes in operational inventory. Diameter classes are labelled by their midpoint, and the class width must be stated. Common errors are measuring at the wrong height, measuring over moss, vines or loose bark, a tape that is twisted or not horizontal, misreading the tape at the overlap, and transcription mistakes. A girth recorded as a diameter (or the reverse) is a factor-of-π error that shows up as an implausible basal area; data checks should flag it.
The Biltmore stick is a quick, low-accuracy alternative: a straight rule held horizontally against the tree at a fixed distance from the eye. Its graduations are calculated so that the stem edges line up with zero and with the mark for the tree's diameter. For a reach (eye-to-stick distance) a and tree diameter D, the mark is placed at D times the square root of a divided by (a + D). It is useful for checks and teaching, not for permanent plots.
Bark is not wood, and timber is sold by volume under bark, so mensuration needs bark thickness. A bark gauge is pushed through the bark to the cambium and the single bark thickness read off; it is measured at breast height, often on two or four sides, and averaged. Diameter under bark is diameter over bark minus twice the single bark thickness, because the bark is crossed twice on a diameter. A tree of 42 cm over bark with bark 1.5 cm thick has 39 cm under bark.
Bark thickness varies by species, age, tree size, position on the stem and site. Thick-barked species such as many pines and fire-adapted savanna trees can carry bark of several centimetres at the base; thin-barked species such as beech have very little. Bark percentage of volume therefore varies widely and must come from local measurements or tables. Inventories commonly predict under-bark diameter from over-bark diameter by a regression fitted to bark-gauge data. Bark thickness is also an ecological trait: it protects the cambium from fire, and studies of fire tolerance use it.
The crown is the photosynthetic engine of a tree, and crown size predicts growth, vigour and response to thinning. Crown width (crown diameter) is measured as the horizontal spread of the crown, usually as the average of two perpendicular measurements from drip-line to drip-line taken by projecting the crown edge vertically to the ground. Crown projection area follows by treating the crown as a circle or ellipse. Crown length is total height minus height to the base of the live crown. Crown ratio, crown length divided by total height, is a widely used vigour indicator: a tree 24 m tall with its live crown starting at 15 m has a crown length of 9 m and a crown ratio of 0.375, or 37.5%.
Crown measurements feed competition indices, crown competition factor and the interpretation of aerial imagery, where crown diameter is used to predict stem diameter. Crown class (dominant, codominant, intermediate, suppressed) is a visual rating of the crown's position in the canopy and is treated in the silviculture subjects of this track.
Several heights are used. Total height is the vertical distance from ground level to the tip of the leader. Bole height or clear-bole height runs to the first major branch or the base of the live crown. Merchantable height runs to a top diameter limit (for example a minimum top diameter set by the market) or to a point where defects stop use. Height to crown base is needed for crown ratio. A question or volume table must say which height it means.
Height is harder to measure than diameter. The top of a tall tree is often hidden, broadleaved crowns have no single tip, and the measurer must stand at a distance. Height is therefore measured on a subsample of trees on many inventories and predicted for the rest from a height-diameter curve.
Most hypsometers use angles. The measurer stands at a known horizontal distance D from the tree, sights the top at an angle α above the horizontal and the base at an angle β. On level ground with the eye above the base, the height is D(tan α + tan β). If both top and base are above eye level, as when standing below the tree on a slope, the base reading is subtracted: height = D(tan α − tan β). In general, add the readings when they lie on opposite sides of the horizontal and subtract when they lie on the same side.
Many clinometers read percent slope, which is 100 tan θ, so the formula becomes height = D × (top percent − base percent) / 100, with readings below the horizontal taken as negative. A reading of +75% to the top and −8% to the base at 20 m gives 20 × (75 + 8) / 100 = 16.6 m. Some instruments carry a topographic scale (in units of 66 ft) or scales for fixed distances such as 15, 20 and 30 m, which read height directly when the measurer stands at that distance.
The distance must be horizontal. If a slope distance S is taped along ground inclined at θ degrees, the horizontal distance is S cos θ. Using the slope distance in place of the horizontal distance overestimates height.
Instruments based on similar triangles need no angle. In one classic form the measurer holds a scale so that its ends align with the top and base of the tree while a pole of known length stands against the trunk; the point where the pole's top appears on the scale gives the height. Another uses a straight stick held at a fixed distance from the eye. These instruments are simple and cheap but are less accurate for tall trees. The principle is still worth knowing because examination questions ask which principle a named instrument uses.
Clinometers with degree and percent scales, the Abney level, and laser and ultrasonic hypsometers that measure distance and angle electronically are in wide use. Airborne LiDAR now estimates the heights of dominant trees over large areas.
The main field errors are: sighting a branch in front of the tree instead of the true top, which in broad crowns overestimates height because the sighted point is closer than the trunk; standing too close, which makes the top-angle large and small angle errors produce large height errors (a distance roughly equal to the tree height is a good rule); not correcting for slope distance; and leaning trees. A tree leaning towards the observer appears taller than it is and one leaning away appears shorter; measuring from a position at right angles to the lean reduces the error.
Basal area is the cross-sectional area of a stem at breast height, normally over bark. Treating the section as a circle, g = π d² / 4. With d in metres the answer is in square metres; with d in centimetres, g in m² = π d² / 40 000 = 0.00007854 d². A tree of 40 cm diameter has g = 0.1257 m². In US units, basal area in square feet = 0.005454 D² with D in inches; a 12-inch tree has 0.785 ft².
Stand basal area, G, is the sum of the basal areas of all trees on a plot, expanded to a hectare (or acre) by the plot's expansion factor, the number of plots of that size in a hectare. A 0.05 ha plot has an expansion factor of 20; a 400 m² plot has 25. If the trees on a 0.02 ha plot have diameters of 20, 25, 30 and 35 cm, their basal areas sum to 0.2474 m², and G = 0.2474 × 50 = 12.4 m²/ha.
Two common slips: using the diameter as if it were the radius in π r², which quadruples the answer, and forgetting to convert centimetres to metres before squaring, which multiplies the answer by 10 000.
The quadratic mean diameter (Dq or Dg) is the diameter of the tree of mean basal area. With G in m²/ha and N stems per hectare, Dq = √(4G / (π N)), in metres. A stand with 20 m²/ha on 400 stems/ha has Dq = 0.252 m or 25.2 cm. Dq is always at least as large as the arithmetic mean diameter, because squaring gives more weight to big trees. It links stand basal area and density exactly (G = N π Dq² / 4) and is the mean diameter used in stand density indices and most yield tables.
Basal area is the forester's most useful single measure of stand density. It is quick to measure, especially with angle gauges, it combines number and size of trees, and it is closely related to stand volume, since volume per hectare is basal area times a mean height times a form factor. It is used to set thinning prescriptions (thin from 30 to 20 m²/ha, for example), to describe stocking relative to a target or a normal yield table, to compare sites and treatments, and as a predictor in growth models. Basal area growth of individual trees is also a better measure of their vigour than diameter growth, because the same diameter increment represents more wood on a large tree than on a small one.
Basal area has limits as a density measure: the same G can be many small trees or a few large ones, which differ in competition and growth. Stand density index and relative spacing, introduced in the growth-and-yield subject, combine N and Dq to address this.
A tree stem is not a cylinder; its taper changes along its length. Mensuration describes stem shape by comparing segments with four solids of revolution: the cylinder, the paraboloid, the cone and the neiloid. In the paraboloid the cross-sectional area falls in proportion to distance from the tip; in the cone it falls with the square of that distance; in the neiloid with the cube. The middle part of a typical stem is close to a paraboloid frustum, the upper part close to a cone, and the flared butt close to a neiloid frustum. Log volume formulas are exact for some of these shapes and biased for others.
With L the log length and g_b, g_m and g_t the cross-sectional areas at the base (large end), middle and top (small end):
Huber's formula: V = g_m × L. It needs only the mid-diameter and length.
Smalian's formula: V = (g_b + g_t) / 2 × L. It needs the two end diameters, which are easy to measure on felled or stacked logs.
Newton's formula: V = (g_b + 4 g_m + g_t) / 6 × L. It needs all three measurements.
For a log 6 m long with diameters of 40 cm at the base, 34 cm in the middle and 30 cm at the top, the cross-sectional areas are 0.1257, 0.0908 and 0.0707 m². Huber gives 0.545 m³, Smalian 0.589 m³ and Newton 0.560 m³.
Newton's formula is exact for all four solids (and their frustums); it is Simpson's rule applied to cross-sectional area. Huber's and Smalian's are both exact for a cylinder and for a paraboloid frustum. For a cone frustum, Huber underestimates and Smalian overestimates, and Smalian's error is twice Huber's in size. For a neiloid frustum such as a butt log, both errors are larger: Smalian overestimates badly. Note also that Smalian's formula averages areas, not diameters; computing the area of the mean of the two end diameters gives a smaller answer (0.577 m³ in the example), which is a different and generally inaccurate estimate.
Smalian's formula is the most used in timber trade and scaling because log ends are visible on a landing or in a stack. Its tendency to overestimate the volume of butt logs is reduced by measuring short logs or sections. Huber's formula is preferred for logs measured in the forest before bucking, or in research with short sections, but the mid-point of a log in a pile is often inaccessible, and bark can be thicker at mid-length on some species. Newton's formula is the most accurate but requires three measurements and is used mainly in research and for checking other methods. For sectional measurement of standing or felled trees in volume-table construction, the stem is divided into short sections (commonly 1–2 m) and each section's volume is computed by Huber or Smalian; the top is treated as a cone (volume = g × l / 3) and the stump is excluded or measured separately.
Bark matters: if volume is wanted under bark, under-bark diameters are used. Commercial rules may use other conventions, such as measuring the small end only, with defined length allowances and diameter rounding. Log scales (board-foot rules in North America, for example) estimate sawn output rather than true solid volume and are a separate topic.
In parts of the Commonwealth, including historically in India and the United Kingdom, round timber was sold by the quarter-girth or Hoppus rule: V = (G/4)² × L, where G is the mid-girth. Because the true cross-sectional area from girth is G² / (4π), the Hoppus volume is π/4, about 0.785, of the true Huber volume. The roughly 21.5% shortfall was a deliberate allowance for waste in squaring the log. The Hoppus cubic foot equals 1.273 true cubic feet. For a log of mid-girth 1.2 m and length 4 m, the Hoppus volume is 0.360 m³ and the true Huber volume is 0.458 m³. A question that mentions quarter girth expects the Hoppus rule; a question that asks for true volume expects π.
The volume of a standing tree cannot be measured directly without climbing or felling, so it is estimated from diameter and height using a form factor. A form factor is the ratio of the tree's volume to the volume of a reference cylinder. The reference cylinder's cross-section and height define which form factor is meant.
The artificial or breast-height form factor, the one most used, takes a cylinder with the basal area at breast height and the total height of the tree: f = V / (g × h). A tree of 0.90 m³ with dbh 35 cm and height 22 m has f = 0.90 / (0.0962 × 22) = 0.425. Conversely, a tree of dbh 30 cm, height 20 m and form factor 0.45 has volume 0.0707 × 20 × 0.45 = 0.636 m³.
The absolute form factor uses the basal area at breast height but only the volume and height above breast height. The normal or true form factor uses a diameter taken at a fixed proportion of total height (for example one-tenth of the height) instead of at breast height; it has the advantage of being independent of tree size, but is rarely used in practice.
For the geometric solids, measured from the base of the solid, the form factor of a cylinder is 1, a paraboloid 1/2, a cone 1/3 and a neiloid 1/4. Real trees have breast-height form factors commonly in the range of about 0.4 to 0.6 for total stem volume over bark, but the value depends on species, age, crown length, stand density and whether volume is total or merchantable and over or under bark. A form factor must therefore be used with the same definitions as when it was derived.
A form quotient is the ratio of a diameter higher up the stem to dbh. It describes taper: the nearer to 1, the more cylindrical the stem. Schiffel's form quotient uses the diameter at half the total height; Jonsson's absolute form quotient uses the diameter at half the height above breast height. A tree of dbh 32 cm with a diameter of 22 cm at half height has a form quotient of 0.69.
Girard's form class, used in North American volume tables, is the diameter inside bark at the top of the first standard 16-ft log (at 17.3 ft above ground, allowing for the stump) divided by dbh over bark, expressed as a percentage. Form-class volume tables are entered with dbh, merchantable height and form class.
Form quotient and form factor are related: trees with higher form quotients have higher form factors, and empirical formulas link the two, but their coefficients differ by species and region. Stem taper equations, which predict diameter at any height from dbh and total height, have largely replaced form quotients in modern work, because they give volume to any top diameter and log assortment by integration.
A volume table gives the average volume of trees of given dimensions. It is built from careful measurements of sample trees and then applied to many trees measured only for diameter (and perhaps height). Three kinds are distinguished by the number of entries.
A local or one-entry table gives volume by dbh alone. It is cheap to use but valid only where the height-diameter relationship is the same as in the data used to build it, typically one species in one district or forest type.
A general or standard table, the two-entry table, gives volume by dbh and height. It is valid over a wider area because differences in height-diameter relations between sites are absorbed by the height entry.
A form-class table adds a third entry, form class or form quotient, to capture differences in taper. It is the most general but needs an upper-stem measurement.
A local table can be derived from a general table by fitting a height-diameter curve for the local stand and reading the general table along it. Tables also differ in what they report: total stem or merchantable volume, over or under bark, to a fixed top diameter, in cubic metres or in trade units. The heading of a table must say which.
Building a volume table or equation follows a standard sequence. A sample of trees, covering the full range of diameters and heights to be used, is selected; typically from several dozen to a few hundred trees per species and region. Each tree is felled (or measured standing with optical instruments or terrestrial laser scanning), and its volume is computed by sections using Huber's or Smalian's formula, with the top as a cone. A regression model is fitted to volume as a function of diameter and height.
Common model forms are: the constant form-factor equation V = b D²H; the combined-variable equation V = a + b D²H (Spurr); and the logarithmic equation ln V = a + b ln D + c ln H (Schumacher-Hall), which is fitted on the log scale. Because tree volume varies more among large trees than small ones, ordinary least squares on the untransformed scale is often replaced by weighted regression, or by the logarithmic form. Back-transforming a logarithmic model gives a median rather than a mean, and a correction factor (exp of half the residual variance) is often applied.
With a combined-variable equation V = 0.02 + 0.000035 D²H, D in centimetres and H in metres, a tree of 30 cm and 22 m has V = 0.02 + 0.000035 × 19 800 = 0.713 m³.
A volume table should be tested on independent trees that were not used to build it. The test checks bias (average difference between table and measured volume) and precision across diameter classes. Tables are used only within the range of data used to build them; extrapolating to trees larger than the sample is a common error. Applying a table from another region or another silvicultural system (for example a natural-forest table to a fast-grown plantation) can produce large bias, because form differs.
Stand volume is obtained by applying the table to each tree, or to each diameter class in a stand table, and summing. In inventory, the error of the volume table adds to the sampling error of the inventory. In India the Forest Research Institute and the Forest Survey of India have produced tables for major species.
Volume serves timber; biomass serves energy, nutrient budgets and, above all, carbon accounting. Biomass is the dry mass of living plant material, usually split into above-ground biomass (stem, branches, foliage) and below-ground biomass (roots). Carbon stock is biomass multiplied by a carbon fraction, and carbon dioxide equivalent is carbon multiplied by 44/12, the ratio of the molecular mass of CO2 to the atomic mass of carbon.
Allometry is the relationship between the size of one part of an organism and another. A biomass allometric equation predicts tree biomass from easily measured variables, most often dbh, sometimes with height and wood density. The commonest form is the power function B = a D^b, fitted as ln B = ln a + b ln D. Exponents for above-ground biomass on diameter are commonly between 2 and 3. With a = 0.1 and b = 2.4, a tree of 30 cm dbh has B = 0.1 × 30^2.4, about 353 kg.
Equations are built by destructive sampling: trees are felled, separated into components, weighed fresh in the field, and subsamples are oven-dried to convert fresh to dry mass. Roots are excavated for a smaller number of trees because it is costly. As with volume equations, the logarithmic fit must be back-transformed with a correction for bias, commonly exp(SEE² / 2) where SEE is the standard error of estimate on the log scale.
Equations are species- and site-specific to a degree. Generic multispecies equations exist for tropical forests that use dbh, height and wood density together; they are widely used when no local equation exists, but local validation is recommended. The choice of allometric equation is often the largest single source of error in a forest carbon estimate.
When volume is known (from inventory and volume tables), biomass can be estimated as volume × basic wood density × biomass expansion factor. Basic wood density is oven-dry mass per green volume, in t/m³; it varies widely by species, roughly from about 0.3 to over 0.9. A biomass expansion factor (BEF) expands stem biomass to whole above-ground biomass, adding branches and leaves; it is larger for young stands and small trees and approaches values a little above 1 for large, mature stems. The IPCC guidelines also use combined biomass conversion and expansion factors (BCEF) that convert volume directly to above-ground biomass. For a stand of 150 m³/ha, density 0.55 t/m³ and BEF 1.3, above-ground biomass is 150 × 0.55 × 1.3 = 107 t/ha.
Below-ground biomass is usually estimated with a root-to-shoot ratio. With a ratio of 0.24 and above-ground biomass of 100 t/ha, roots add 24 t/ha. Default ratios are tabulated by ecological zone and above-ground biomass class in the IPCC guidelines; local values are preferred.
The carbon fraction of dry wood is close to one half, and two conventions coexist. Many studies and older guidance use 0.5. The IPCC 2006 Guidelines give a default of 0.47 for above-ground forest biomass, with ranges by component and climate. National inventories and carbon standards specify which to use. A tree of 500 kg dry biomass holds 235 kg of carbon at a fraction of 0.47 (250 kg at 0.5), equivalent to about 862 kg of CO2 at 0.47. Dividing by 44/12 instead of multiplying is a common slip that gives a number smaller than the carbon itself.
A stand is described from measurements on sample plots. A fixed-area plot (circular, square or rectangular) of known area is laid out and every tree above a minimum diameter is measured. Each tree then represents as many trees per hectare as there are plots of that size in a hectare. A circular plot of radius 11.28 m has an area of about 400 m², so each tree represents 25 trees per hectare; 14 trees on such a plot means about 350 stems per hectare. Circular plots of radii 5.64, 7.98, 11.28, 12.62 and 17.84 m correspond to 100, 200, 400, 500 and 1000 m².
Trees on the boundary are in if the centre of the stem at breast height is inside the plot. On slopes, plot radius is measured horizontally, or slope distances are corrected; otherwise the plot is smaller than intended and stocking is underestimated. Plot design and the number of plots are covered in the forest-inventory-sampling subject.
A stand table lists stems per hectare by diameter class, sometimes by species; a stock table adds basal area or volume per class. From the stand table come stems per hectare (N), basal area per hectare (G), quadratic mean diameter (Dq), and the diameter distribution, which in even-aged stands is roughly bell-shaped and in uneven-aged stands declines steeply from small to large classes (a reverse-J shape).
Mean height can be defined in several ways. The arithmetic mean height of all trees is rarely used. Lorey's mean height weights each tree's height by its basal area: H_L = Σ(g h) / Σg. It is the natural height to multiply by G and a form factor, and it is less affected by the many small trees in a stand. For trees of 20, 30 and 40 cm with heights 15, 20 and 24 m, the arithmetic mean height is 19.7 m and Lorey's height is 21.5 m.
Top height describes the tallest part of the stand and is used for site assessment because it is little affected by thinning from below, which removes small trees. There are several definitions, and they give different numbers. A common one, used widely in Europe and in many yield tables, is the mean height of the 100 largest-diameter trees per hectare; on a 0.04 ha plot that means measuring the 4 largest trees. Others use the 100 tallest trees per hectare, the mean height of dominant and codominant trees, or the height of the tree of mean basal area of the largest 20% of stems. Indian yield tables traditionally used the mean height of the dominant trees (or the top height defined in the table's introduction). Always use the definition that belongs to the table or site-index curves being applied.
Stand volume per hectare can be obtained in three ways. The most accurate is to apply a volume table or equation to every measured tree and expand. A quicker method multiplies basal area per hectare by a mean height (often Lorey's) and a stand form factor: a stand of 24 m²/ha with a mean height of 18 m and a form factor of 0.45 has about 194 m³/ha. The third uses stand volume equations that predict volume per hectare from G and top height directly. Stand volume can also be read from yield tables if age and site quality are known, which is an estimate of what a normally stocked stand of that site should carry rather than a measurement.
Angle-count sampling, devised by Walter Bitterlich in Austria in the 1940s and called point sampling, variable-radius plot sampling or prism cruising, measures basal area per hectare without measuring any plot. The observer stands at a sample point and turns through 360°, sighting every tree at breast height through an angle gauge that projects a fixed horizontal angle. A tree whose stem appears wider than the angle is counted ('in'); a tree that appears narrower is not ('out'); a tree that appears exactly equal is borderline. Basal area per hectare is the count multiplied by the basal area factor (BAF) of the gauge.
The method works because each tree has its own circular plot whose radius is proportional to its diameter. A tree is counted if the observer stands inside that circle. Large trees are counted from farther away, so trees are sampled with probability proportional to their basal area, and each counted tree contributes exactly the same basal area per hectare: the BAF.
For a gauge with a horizontal angle α, the BAF in m²/ha is 10 000 sin²(α/2). For a stick gauge with a crosspiece of width w held at distance L from the eye, sin(α/2) is close to w / 2L, so BAF = 2500 (w / L)². A 2-cm crosspiece at 50 cm gives BAF = 2500 × (0.04)² = 4 m²/ha. Common metric BAFs are 1, 2 and 4 m²/ha; in the United States BAFs are in ft²/acre, 10 and 20 being common.
The limiting distance for a tree of diameter d is the farthest the tree can be from the point and still be counted: R = d / (2 sin(α/2)). In metric units with d in centimetres and R in metres, R = d / (2√BAF), or d multiplied by a plot radius factor. For BAF 2, a 30-cm tree has a limiting distance of 10.6 m; if it stands 11 m away it is out. Borderline trees are checked by measuring the distance and diameter; if there is no time to check, the convention is to count every second borderline tree, or to count each as half.
Basal area per hectare at a point is the number of in-trees times the BAF. With BAF 4, seven clear in-trees and two borderline trees counted as half each give (7 + 1) × 4 = 32 m²/ha. The average over several points estimates stand basal area.
Each in-tree also represents a number of stems per hectare equal to BAF divided by its own basal area. With BAF 4, a counted tree of 25 cm (g = 0.0491 m²) represents 81.5 stems per hectare; a counted tree of 50 cm represents about 20. Multiplying each tree's per-hectare representation by its volume and summing gives volume per hectare; a simpler route multiplies each tree's volume-to-basal-area ratio (its 'VBAR') by the BAF.
Angle gauges include the simple stick gauge, wedge prisms that deflect the image of the stem by a fixed angle (the tree is in if the displaced image still overlaps the real stem), and relascopes, which are optical instruments with several angle bands and automatic correction for slope. A wedge prism must be held over the sample point, not over the observer's eye at an offset, and rotated about the point.
Errors come from: hidden trees behind other trees (the observer must step sideways to check, keeping the same distance); mis-judged borderline trees; slope (on a slope the horizontal angle is effectively smaller, and the gauge or count must be corrected, or a self-correcting relascope used); leaning trees; and choosing a BAF that gives very few or very many trees per point. A BAF that gives roughly 5 to 12 in-trees per point is a common rule of thumb. Larger BAFs suit stands of large trees, smaller BAFs suit open or small-tree stands.
In temperate and boreal trees, and in tropical trees with a distinct dry or cold season, the cambium lays down one growth ring a year. Counting rings on a stump or on a core extracted with an increment borer gives age at the height of the count. To get total age, the number of years the tree took to reach that height is added; that number comes from local data on young trees. A tree with 42 rings at breast height, of a species that takes about 6 years to reach breast height on that site, is about 48 years old.
Ring counts can mislead. False rings form when growth stops and restarts within a season, for example after drought or defoliation. Rings can be missing in suppressed trees or in very bad years, especially near the base or on one side of the stem. Many tropical trees, particularly in aseasonal wet climates, have no annual rings at all. Cross-dating, matching patterns of wide and narrow rings between trees, is the dendrochronological method that detects false and missing rings. Increment cores should be taken towards the pith; a core that misses the pith undercounts, and the missing rings are estimated from ring curvature.
Plantation records give age directly and are the most reliable source in planted forests. Many conifers, notably pines and spruces, produce one whorl of branches per year in youth, so counting whorls (or branch scars) gives age for young trees. For tropical species without rings, age is estimated from records, from repeated measurement of growth on permanent plots, or from radiocarbon dating in research.
An increment borer extracts a thin core from bark to pith. Besides age, it gives recent radial growth: the width of the last n rings. Diameter growth inside bark over those years is twice the radial growth, because the radius grows on both sides of the pith. If the last 10 rings span 25 mm of radius, the tree grew about 50 mm in diameter in 10 years, or 5 mm a year inside bark. Converting to over-bark growth needs the bark ratio. Core-based growth is used in temporary-plot inventories where remeasurement is not possible.
Stem analysis reconstructs the complete growth history of a felled tree. Discs are cut at the stump, at breast height and at regular intervals up the stem. On each disc the rings are counted and the radii to the end of each age period (for example every 5 or 10 years) are measured along two or four directions.
The ring count on each disc gives the age at which the tree reached that height: tree age minus the number of rings at that height. A 30-year-old tree with 24 rings on a disc at 3.3 m reached 3.3 m at about age 6. Plotting the heights of the discs against these ages gives the height-age curve, which is the basis for site-index curves; refined methods interpolate the true height at each age within a section. The radii give diameter at each height at each past age, from which the stem profile and volume at past ages are reconstructed and diameter, basal area and volume increment curves are drawn. Stem analysis is laborious but gives growth series that would otherwise need decades of remeasurement.
Growth is the increase in size over time; increment is the amount of growth in a stated period. Increment can be measured in diameter, basal area, height or volume, for a tree or a stand. Three expressions of increment are used.
Current annual increment (CAI) is the growth in one particular year.
Periodic annual increment (PAI) is the growth over a period of several years divided by the number of years: (V2 − V1) / n. Because annual remeasurement is rarely practical, PAI over 5 or 10 years is used in place of CAI.
Mean annual increment (MAI) is the total size at a given age divided by that age: V_t / t. It is the average rate of production from establishment to that age.
A stand of 165 m³/ha at 30 years and 240 m³/ha at 40 years has a PAI of 7.5 m³/ha/yr between those ages and an MAI of 6.0 m³/ha/yr at age 40.
Growth of an even-aged stand follows a sigmoid (S-shaped) cumulative curve: slow at first, then rapid, then slowing as the stand matures. CAI rises to a peak early and then falls. MAI rises more slowly, peaks later, and falls more slowly. The two curves cross at the peak of MAI: while CAI is above MAI, each additional year raises the average, and once CAI falls below MAI, the average starts to decline. So MAI is at its maximum when CAI equals MAI.
This has a direct management use. The age at which MAI culminates is the rotation that maximises the average volume produced per hectare per year, called the rotation of maximum volume production or the technical rotation for volume. Financial, silvicultural and technical rotations are usually different and are treated in the forest-management subject. Reading a yield table: compute MAI for each age and find its peak. For volumes of 80, 115, 150, 178, 200 and 216 m³/ha at 20, 25, 30, 35, 40 and 45 years, MAI is 4.0, 4.6, 5.0, 5.09, 5.0 and 4.8; it culminates at 35 years, while the five-year PAI peaked earlier, between 20 and 30 years. Faster-growing species and better sites reach both peaks earlier.
Increment is often expressed as a percentage of the size of the tree or stand. Different formulas give slightly different results for the same data, so a question will name the formula.
Simple interest on the initial value: p = 100 (V2 − V1) / (n V1).
Pressler's formula, which uses the mean of the initial and final values: p = (200 / n) × (V2 − V1) / (V2 + V1).
Compound interest: p = 100 [(V2 / V1)^(1/n) − 1].
For a tree that grew from 0.80 to 1.00 m³ in 5 years, these give 5.0%, 4.44% and 4.56%. Increment percent declines with age and size even while absolute increment is still high, because the base keeps growing. It is used in older methods of yield regulation and to compare the growth of trees of different size. Other classical rules estimate volume increment percent from the number of rings in the outer centimetre of radius and the diameter, with a constant that depends on crown development; they are approximate and need local calibration.
Growth is measured by remeasuring permanent sample plots, which is the most reliable method and the basis of growth-and-yield modelling; by increment cores and stem analysis, which reconstruct past growth; and by dendrometer bands, which record diameter change over weeks or seasons. Apparent negative growth from measurement error must be checked, not deleted blindly. Stand growth has components: survivor growth, ingrowth (trees crossing the minimum diameter), mortality and harvest. Gross growth includes mortality, net growth subtracts it, and their definitions must be stated in any growth report.
Site quality is the productive capacity of a site for a particular species or forest type. It depends on climate, soil, topography and their interactions, and it is the main reason why stands of the same species and age carry different volumes. Measuring it directly, as volume production over a rotation, would take decades, so foresters use indirect measures. The commonest is site index: the average height of the dominant (or top-height) trees at a reference age, called the base or index age.
Height of dominant trees is used because it depends strongly on site and only weakly on stand density over the normal range, and it is not changed much by thinning from below. Volume and diameter, by contrast, depend heavily on density and treatment.
The base age differs by species and country. In North America, 50 years is common for many natural-forest species and 25 years for fast-growing plantation species; plantation species with short rotations use shorter base ages still. A site index of 25 m at base age 50 means the dominant trees are expected to be 25 m tall at age 50. Some systems, including many older European and Indian yield tables, group sites into quality classes instead of a continuous index: Site Quality I, II, III and so on, where I is the best site in the Indian and British convention. The classes are defined by ranges of top height at given ages in the yield table. A question must state the base age and the convention.
Site-index curves show height against age for each site index. The guide-curve method fits an average height-age curve to data from many stands and then draws the other curves proportional to it; these are anamorphic curves, which have the same shape and differ by a constant ratio. To estimate site index with an anamorphic system, multiply the measured height by the ratio of the guide-curve height at base age to the guide-curve height at the stand's age. If the guide curve gives 15 m at 30 years and 21 m at the base age of 50, a stand of top height 18 m at 30 years has a site index of 18 × 21 / 15 = 25.2 m.
Real height growth patterns often differ in shape between good and poor sites: good sites may grow fast early and slow down sooner. Polymorphic curves, built from stem analysis or remeasured permanent plots, allow the shape to vary with site. Curves fitted from temporary plots of different ages can be biased if good and poor sites are not equally represented at all ages. Measuring site index in a very young stand is unreliable because early height growth is sensitive to establishment conditions; growth-intercept methods, which use the height growth over a fixed number of years above breast height, are an alternative for young stands.
Where suitable trees are absent, site is assessed by other means. Soil-site studies relate site index to soil depth, texture, drainage, topographic position and climate, and predict it for unforested land. Indicator plants, ground vegetation species that reflect site conditions, are used in some forest typing and site classification systems (well developed in Finland and in parts of Canada and Germany). Combined ecological land-classification systems integrate climate, soil and vegetation. Site index is not suitable for uneven-aged or mixed stands, where trees of the same height may be of very different ages and suppressed for decades; there, height-diameter relationships of dominant trees or soil-based methods are used. Site quality assessment itself is treated more fully in the silviculture-principles subject; this chapter covers the measurement side.
Learn the twenty or so formulas in this subject until they are automatic, with units: girth to diameter, basal area, expansion factors, the three log formulas, form-factor volume, the trigonometric and percent height formulas, BAF and limiting distance, MAI, PAI and the three increment-percent formulas, and anamorphic site index. In practical classes, measure the same trees with tape and calipers, and the same tree from two distances, and see how the errors behave. In examinations, write the formula first, convert centimetres to metres before squaring, and state any convention you assume (breast height, carbon fraction, top-height definition). Descriptive answers score well when they explain why a method works, for example why angle-count sampling gives basal area directly.
A forester five years into practice uses mensuration daily: setting up and remeasuring plots, checking crews' data for girth-diameter mix-ups and impossible heights, choosing a BAF for a cruise, applying the right volume table to a coupe, estimating under-bark volume for a sale, and converting inventory volume to biomass and carbon for a project report. The key decisions are which table or equation is valid for the stand, how many trees to measure for height, how to handle buttressed or forked trees consistently, and whether an apparent change between measurements is real growth or measurement error. Good practice is to document conventions on every data sheet and to keep a small set of checked local equations.
A senior forester relies on mensuration indirectly: growing-stock estimates feed the allowable cut, increment figures justify yields, and carbon inventories support climate reporting and payments. The decisions at this level are about data quality and method: commissioning new volume and biomass equations when plantations or climates change, insisting on permanent-plot networks that survive staff turnover, standardising conventions across divisions, and judging whether an estimate is precise enough for the decision it supports. Mensuration errors at this scale can mean over-cutting or lost credibility in carbon markets.
A senior researcher works at the frontier of measurement: terrestrial and airborne laser scanning for non-destructive volume and biomass, taper and biomass models that include wood density and crown architecture, error propagation from tree measurement through allometry to national carbon totals, and long-term growth series from permanent plots and tree rings under climate change. Such a researcher also trains the next generation and sets examination questions, and needs to keep classical methods (Huber, Smalian, Bitterlich, stem analysis) connected to modern data so that students understand what the new tools estimate.
This manual does not reproduce any national volume table, yield table or allometric equation; the coefficients used in examples are illustrative, and real work must use the equations validated for the species and region. It gives no default wood densities, root-to-shoot ratios or biomass expansion factors beyond illustrative values; use the IPCC tables or national data. Board-foot log rules, North American log scaling and defect deduction, detailed stem taper equations, terrestrial laser scanning methods and the statistics of error propagation are only mentioned. Rules of thumb that estimate increment percent from ring counts (such as Schneider's) are described without constants because textbook constants vary. Sampling design for inventories is in the forest-inventory-sampling subject and growth models in growth-yield-modelling.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions.
These notes are free to read here, with no account.
What the Laisha app adds:
Free for ten days, then a subscription. We would rather write that here than have you find it later.