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The Popular Concept of Tree Size Predicted by Radius of the Trunk

John J. Sabuco

Good Earth Company
Flossmoor, IL

A paper from the Proceedings of the 9th Metropolitan Tree Improvement Alliance (METRIA) Conference held in Columbus, Ohio, August 8-10, 1996.


It is difficult to establish the value of a lost tree when negligence, carelessness or vandalism precipitates its death. Though there have been numerous algorithms available for the estimation of above-ground biomass of wild tree species, these algorithms are species-specific equations and their aim is the clarification of ecological data (Cain & Castro 1959, Mueller-Dombois & Ellenberg 1974), not the satisfaction of a consumer. The ecological view of a tree as biomass is valid to a point (Greig-Smith 1984), but many or most consumers see the size of a tree as the size of the crown and the heft of the trunk. Furthermore, the "size" of a crown, according to the popular concept, is its breadth and height as well as the even spacing of its branches, not necessarily the density of the crown and certainly not the weight of its wood.

It becomes immediately obvious how this can become an issue of concern when one considers that the wood of Quercus alba, the white oak, has twice the specific gravity of Taxodium distichum, the bald cypress (46.35 to 28.31 lbs d.w.) (Hough 1947). An owner of a now dead bald cypress might be quite perturbed if a white oak equaling its mass was used as a replacement, resulting in a tree roughly one half of its size. Conversely, the popular measure of basal area (the area of the cross section of a trunk at 1.37 m (4.5 ft) above the ground) cannot be relied upon, for an oak of considerable size might have half the basal area of a bald cypress which is far smaller in size, due to the significant buttress of a cypress' trunk. Even trees more closely related in natural habitat than oak and cypress have significant differences between basal area and size. Consider that Carya ovata, the shagbark hickory (52.17 lb d.w.), of equal height and breadth to nearly any oak with which it might share habitat, will be of greater age with a far more slender trunk (Hough 1947). It is of no value to the public to use such measures to settle disputes regarding tree loss since these measures are not applicable to the popular concept of tree size, and one party or the other will likely remain disgruntled.

It is the purpose of this paper to establish an equation for the rate of growth for trees based on the popular concept of tree size, that can be used to establish a ratio of value from the known value of nursery-grown trees and trees of sizes greater than those that can be found in a nursery.


Gathering data on a variety of large trees

The model of tree size used is the volume contained in the outline of the edge of the canopy and the volume of the trunk. Trees numbering 337 were selected at random by using a random numbers chart printed in Snedecor (1989) in a manor prescribed by Greig-Smith (1984). Trees were located in 17 woodland and open habitats in northeastern Illinois and northwestern Indiana. South Bend is the nearest climatological station to Reynolds, Indiana. The range of sites represents a dynamic range in climate (NOAA 1983, USDA 1941) and soil (USDA 1938) conditions from Illinois to Indiana: 33.34" to 36.09" of rainfall per year, 42 to 72.2" of snowfall and the soils change from clay glacial till to sand and sandy loam. Chicago has an average of 16 days over 90°F. and South Bend has 10 days over. Chicago averages 14 days below 0°F. and South Bend averages 8 such days. Average wind speeds are identical.

Multi-stemmed specimens were rejected, but senescent trees were not eliminated. The heights of various points on the trees were determined by use of a Sylvan Clinomaster climometer. The breadth of the crown was determined by dual lateral observations at 90o angles from a tape drawn across the ground below the tree. The crown was classified according to defined geometric shapes - in combination if necessary. Those shapes are:

The dbh of the trunk was determined with the use of a tape. Significant root flares or buttresses were also accounted for.

Once the trees' volumes were determined, the data were used to create a scatter-plot with radius plotted on the ordinate (x axis) and volume on the abscissa (y axis) of the graph. The data describe an exponential curve and the best fit line was plotted as well as a 95% confidence ellipse. Given the constraints of the total scattergram, the ellipse represents a range of data within which it is 95% certain that the best description of y is the equation of the best fit curve. The species encountered were:

Scientific Name Common Name
Acer rubrum Red Maple
Acer saccharinum Silver Maple
Acer saccharum Sugar Maple
Carya ovata Shagbark Hickory
Crataegus mollis Downy Hawthorn
Crataegus pruinosa Frosted Hawthorn
Fraxinus americana American Ash
Fraxinus pennsylvanica Green Ash
Fraxinus pennsylvanica subintegerrima Red Ash
Juglans nigra Black Walnut
Juglans cinerea Butternut
Prunus americana American Plum
Prunus serotina Black Cherry
Prunus virginiana Chokecherry
Quercus alba White Oak
Quercus ellipsoidalis Shingle Oak
Quercus X hawkinsae Hawkins' Oak
Quercus macrocarpa Bur Oak
Quercus X paleolithicola Hill's Oak/Black Oak hybrid
Quercus palustris Pin Oak
Quercus rubra Red Oak
Quercus X schuettii Swamp White/Bur Oak hybrid
Quercus velutina Black Oak
Tilia americana American Linden
Ulmus americana American Elm
Ulmus rubra Slippery Elm

Collection of data for small trees

150 trees were selected at random points within nursery rows at Teerling Nursery Company, Lockport, Illinois. Ten individuals in three trunk caliper classes: 2", 3", & 4" were measured as before. The trees were divided as follows:

Number Taxon
30 Acer rubrum 'Red Sunset'
30 Fraxinus X excelsior 'Patmore'
30 Fraxinus americana 'Autumn Blaze'
30 Acer nigrum 'Green Mountain'
30 Gleditsia triacanthos 'Skyline'


Data were analyzed using Statistica software (StatSoft 1996). The graphs of both sets of data were analyzed separately and then joined as one data set. A variety of curves were fit to the data and it was found that the exponential curve had the fewest number of data points outside of the 95% confidence ellipse.

Equations were established for all three curves and a simplification of the equation for small trees was determined. Each data point in the scattergram is shown as a relatively large circle. After considerable experimentation, we were able to determine the margin of error for measuring the tree crowns. These circles are indicative of that error.


The large tree data produced only 20 trees outside of the 95% confidence range. That represents 6.3% of that data (22 trees were eliminated from the 337 original trees due to extreme difficulty and disagreement regarding how best to calculate their volumes).

The small tree data produced just five specimens outside the 95% confidence ellipse, representing just 3.3% of the data. No trees were eliminated.

The combined graph and equation has 34 individuals outside the 95% confidence ellipse. This is 7.3% of the data. If the analysis is limited to just those radii equal to or less than 35 cm, then only 15 trees lie outside the 95% confidence ellipse or just 3.2%. Thirty-five cm is a radius of 13.78" or a diameter of 27.55".

The equation for any exponential curve is y=aebx where a is a variable describing the y intercept, b is a variable describing the slope of the curve, e is Euler's constant - the base of all natural algorithm - and x is the radius in this case.

The curves described by the data are: [Note: the expression in parenthesis represents an exponential function]

Small Trees y=218.8 e(0.937x)
Large Trees y=63.209 e(0.085x)
Combined y=6.857 e0.157x)

The simplification for the small tree curve is 1.9er where r is the radius. This simplification is quicker to calculate and is based on the differentiation of the range of radii from 2 cm to infinity, and therefore, it should not be used for radii of less than 2 cm.


Once the volume of a destroyed or damaged tree is calculated, then a proportion is set up that follows the form

Cost of a larger tree = [Volume of larger tree x cost of small tree] / Volume of smaller tree.

The result is a replacement value adjusted for time and growth rate.

A second application of this method is the consistent pricing of nursery stock. A nurseryman can use the simplification of the small tree equation to calculate the volume of small trees, say 2" dbh (5.08 cm), for he is certain of the cost to produce. He then may calculate the volume of the larger tree he wishes to price. The same equation applies to the proportion, resulting in a fair price for the larger tree.

There are four areas of concern or possible improvement in the evaluation of such data.

  1. A measure of crown intensity would be of benefit.
  2. All of the small trees were evaluated at one site. A more heterogeneous sample would have been in order, perhaps spread among nurseries in the same range of conditions as the open land trees were found. Unfortunately, time was critical in the summer of 1996, and activities were curtailed due to horrendous weather conditions. (17" of rain in a single day was just one of the frustrations.)
  3. The estimations of crown diameter and geometric shape were not as precise as the measure of radius; however, the tightness of the data to the curve was indicative of a reasonable degree of accuracy.
  4. As size increases, senescence of trees begins to alter the shape of the curve forming a splined curve which dips at the distal end. This is because canopies lose branches as they begin to deteriorate. The nature of the method is such that considerable deterioration must take place before the result is affected because internal thinning will not impact the overall volume. Nevertheless, one must decide, if a tree of advanced age is more or less valuable than a younger tree, and this is beyond the purview of this method.


The equation presented here, which will be called the tree size illustrator, can be a valuable tool in approximating the popular market-driven view of a tree's size and therefore, by extrapolation, its worth. It is not meant to be an ecological tool in this form, but rather an economic tool. The nurseryman, landscape contractor, arborist, insurance claims adjustor, and the courts may all derive some benefit from its application.


I wish to thank Wendy Erler and Cynthia Lyng for their Herculean effort in the collection of the data and the time-consuming and mundane analysis of the data.


Cain, Stanley A. G.M. de Oliveira Castro 1959 (1971). Manual of Vegetation Analysis. Hafner, New York.

Greig-Smith, Peter. 1984. Quantitative Plant Ecology, 3rd Ed. Butterworths, London; MacMillan, New York.

Hough, Romeyn Beck. 1947. Handbook of the trees of the Northeast US &Canada East of the Rocky Mountains. MacMillan, New York.

Muller-Dombois, Dieter, Heinz Ellenburg. 1974. Aims & Methods of Vegetation Analysis. John Wiley & Son, New York.

National Oceanic & Atmospheric Administration. 1983. Local Climatological Data, Annual Summary with Comparative Data. US Government Printing Office, Washington.

Snedecor, George W. 1989. Statistical Methods, 8th Ed. Iowa State University Press, Iowa City.

United States Department of Agriculture. 1938. Yearbook of Agriculture - Soil & Man. United States Government Printing Office, Washington.

United States Department of Agriculture. 1941. Yearbook of Agriculture - Climate & Man. United States Government Printing Office, Washington.

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