# Getting to Point Sampling

Posted: March 19, 2014

This article was written to help you understand sampling. It is modified from *Plotless Cruising* by O. Lynn Frank, whose explanation of how this system works is one of the best I have read. Sampling is a tool natural resources professionals use to get a sense of the volume (size class and diameter), species, and regeneration (existing and potential) across a landowner’s property. If you’ve ever taken a forestry class or an educational event for forest landowners, perhaps you learned how to assess the volume, species, and quality of trees on a fixed-diameter plot using a Biltmore stick. In that exercise, you, maybe, measured all the trees on a 1/10th acre plot with a 37.2 foot radius. By knowing the plot size, you could expand the volume from your one sample to describe volume per acre. Plots can also provide information about stocking or how crowded the trees are in the stand. To describe stocking, foresters refer to basal area.

Tree DBH (inches) |
Imaginary Circle Radius in Feet (DBH X 33) |
Imaginary Circle Area (acres) |
Number of Trees per Acre (square feet) |
Basal Area of One Tree (square feet) |
Basal Area per Acre (square feet) |
---|---|---|---|---|---|

6 | 16.5 | 0.019635 | 51.02 | 0.196 | 10.0 |

8 | 22.0 | 0.034907 | 28.65 | 0.349 | 10.0 |

10 | 27.5 | 0.054542 | 18.35 | 0.545 | 10.0 |

12 | 33.0 | 0.078540 | 12.74 | 0.785 | 10.0 |

14 | 38.5 | 0.106901 | 9.35 | 1.069 | 10.0 |

16 | 44.0 | 0.139626 | 7.16 | 1.396 | 10.0 |

18 | 49.5 | 0.176715 | 5.66 | 1.767 | 10.0 |

20 | 55.0 | 0.218166 | 4.59 | 2.183 | 10.0 |

22 | 60.5 | 0.263981 | 3.79 | 2.640 | 10.0 |

24 | 66.0 | 0.314159 | 3.18 | 3.142 | 10.0 |

26 | 71.5 | 0.368701 | 2.71 | 3.687 | 10.0 |

28 | 77.0 | 0.427606 | 2.34 | 4.276 | 10.0 |

30 | 82.5 | 0.490874 | 2.04 | 4.909 | 10.0 |

32 | 88.0 | 0.558505 | 1.79 | 5.585 | 10.0 |

34 | 93.5 | 0.630500 | 1.59 | 6.305 | 10.0 |

36 | 104.5 | 0.787580 | 1.27 | 7.876 | 10.0 |

40 | 110.0 | 0.872665 | 1.15 | 8.727 | 10.0 |

The Pencil-size Sapling | |||||

0.28 | 0.773 | 0.000043 | 23,255.81 | 0.00043 | 10.0 |

The 16-foot-diameter Giant Sequoia | |||||

192 | 528.0 | 20.106193 | 0.05 | 201.062 | 10.0 |

As a start to understanding basal area as a measure of crowding, consider that every living tree exerts an influence on its surrounding area. This influence results from the spread of the crown and the root system—and, also from more ephemeral phenomena such as airborne molecules given off by the tree, which can affect nearby trees. (In one study, airborne molecules from a tree under insect attack were shown to incite surrounding trees to take defensive measures against the insects even before they themselves were attacked.) This “biological area of influence” around a tree is irregular, and for all practical purposes, impossible to measure; however, in general, a big tree exerts a bigger influence than a little tree. As well, the big tree’s influence affects things farther away.

A measurable area of influence is obviously needed to serve as a substitute for the tree’s biological area of influence. Such a substitute can be provided by assigning an imaginary circle around the tree proportional to its diameter.

Almost all forestry work relating to timber management involves measuring the sizes of trees. One measurement always taken is the diameter of the tree trunk at a standard height. In the United States, the standard height is 4½ feet above the ground on the uphill side of the tree, and is called breast height. For few of us, 4 ½ feet falls directly at our breast height, so it’s more often a “shoulder” or “nose” height. This is what is referred to as DBH, diameter at breast height.

The cross-sectional area of a tree trunk at breast height is the tree’s basal area. Basal area is usually expressed in square feet rather than square inches because it keeps the numbers smaller. Since we can’t cut the trees down to measure the cross-sectional area, we use formulas to calculate the area. Basal area is 0.005454 times the tree diameter squared. So a twelve-inch tree has a basal area of 0.78 square feet (0.005454 X 122=0.78), and a six-inch tree has a basal area of 0.20 square feet. Therefore, four six-inch tees have the same basal area as one twelve-inch trees.

Foresters use basal area per acre as a measure of crowding in a stand of trees. That measure of crowding is called stocking density. To have a sense of crowding, we have to know two things: basal area per acre and number of trees. The trick is to understand that the basal area is additive and across the stand, and “all” trees – both large and small – contribute to estimating basal area.

It is possible to calculate basal area by measuring diameters of all the trees in the plot. That total, as with volume, can be expanded to a per-acre basis Trees in a stand are generally spaced irregularly, and the stocking density varies from place to place within the stand. To get a good average density of the whole stand, it is necessary to take several plots in the stand and average them.

Basal area and numbers of trees provide a measure of stocking. Neither basal area nor number of trees alone is sufficient to describe stocking. A red oak dominated stand with 140 square feet of basal area would be described as well stocked, but that is not sufficient to “picture” the stand. If the average tree diameter is twelve inches or six inches, the image of the stand is quite different. With a twelve inch average diameter, there are about 180 trees per acre. At six inches, there are about 700 trees.

Trees in a stand are generally spaced irregularly, and the stocking density varies from place to place within the unit. To get a good average density, it is necessary to average data from several plots. The time needed to do this using the fixed-area plot method is considerable, to say the least. Point sampling allows the user to quickly estimate basal area per acre directly by standing at a point to sample trees using either an angle gauge or wedge prism of a known basal area factor (BAF). Neither a plot of known area nor measurement of tree diameters is required.

It is easier to comprehend the concept of point sampling by using the angle gauge. The angle gauge in its simplest form is a card with a notch cut in one edge. The cruiser stands at the point, holds the angle gauge a certain constant distance from his or her eye, and looks at each tree around the point through the notch. When using the angle gauge or card, in this case, your eye remains over of the sample point. Any tree close enough to the point to appear wider than the notch is said to be “in,” and is counted. Any tree that appears narrower than the notch is said to be “out,” and is ignored.

The main practical problem with the angle gauge is the difficulty of consistently holding it the specified distance from your eye. If the distance between your eye and the angle gauge is not correct, error is introduced. Some gauges are mounted on the end of a stick, like the front sight of a rifle; others are equipped with a chain whose end is held in your front teeth. Both make the gauge more bulky and cumbersome to carry through the woods.

The wedge prism eliminates these problems. The prism is a thin wedge of glass with two flat surfaces polished as precisely as a lens to a very specific small angle. The cruiser holds the prism over the point with the flat surfaces vertical and the taper pointing sideways. (Note: the prism is held over the point, which is different from the angle gauge.) The prism will offset a section of the tree image toward the thin end of the prism. The farther away the tree is, the more its view is offset. If the offset section overlaps the direct view over (or under) the prism, the tree is said to be “in,” and is counted. If the view of the tree through the prism does not overlap the direct view, the tree is said to be “out,” and is ignored.

With either instrument, only trees meeting the “in” criteria are counted. The bigger the tree, the further away from the plot center it can be and still be counted. The width of the notch in the angle gauge, and the angle between the two surfaces of the prism are critical, for these determine the basal-area factor of the respective instrument. This explanation will presume BAF of 10 (which is by far the most common factor used in the eastern United States). The BAF describes a ratio between the tree diameter and something called the limiting distance.

If the reader carefully reads the explanation below, then checks with a calculator a few of the diameter classes in the table above, the validity of the system will begin to emerge from the fog of confusion into the clear sunlight of understanding.

Nature distributes trees in a forest in a random fashion, and attempts to make every stand fully stocked right from the start. Therefore, a young stand will contain thousands of trees per acre. Big trees, of course, take up more space than little trees. As the trees grow bigger, they begin to crowd one another in a struggle for survival. The weakest ones die and make room for the stronger ones to grow bigger. As the years go by and the surviving trees get bigger, the struggle continues until eventually only a few dozen big trees occupy most of the space where thousands of little seedlings started out.

But not all the little trees die. Some trees grow more slowly than others, but hang on to life, and continue to occupy small spots in the stand among the bigger trees. They are tough enough to survive, but not vigorous enough to grow large. In a typical undisturbed forest, there are small trees of various sizes among the bigger trees. An average forest has a few big trees and a lot of smaller ones.

The largest diameter class usually has the fewest living trees. As diameter classes increase in size, there are usually progressively fewer trees in the next size class. When all the trees on an acre of undisturbed forest are measured, and a graph of the number of trees in each diameter class is drawn, the points usually suggest a fairly smooth curve that shows an orderly reduction in the number of surviving trees in successively larger diameter classes.

To visualize the relationship between the size of a tree and the number of trees of that size per acre, imagine a circle around every tree proportional to that tree’s diameter. Big trees will have big circles; and little trees will have little circles. The actual size of each imaginary circle is controlled by the basal-area factor of the instrument used. When an angle gauge or prism of “10” basal-area factor is used, the radius of the imaginary circle around any tree is thirty-three times the diameter of that tree. Thus the radius of the circle around a twelve-inch-diameter tree (one-foot diameter) is exactly thirty-three feet. A circle of thirty-three-foot radius has an area of 3,421.17 square feet. This equals 0.0785 acre, or about 1/13 of an acre (more precisely, 1/12.74 acre). In order for the cruiser’s angle gauge or prism to detect a twelve-inch tree, the cruiser must be standing inside that tree’s imaginary circle. The cruiser must be no farther than thirty-three feet from the center of the tree. Since that imaginary circle is about 1/13 of an acre, the assumption is made that there are about thirteen twelve-inch trees on every acre in the stand represented by the sampling point (more precisely, 12.74 such trees per acre). Each twelve-inch tree has a basal area of 0.785 square feet, so these 12.74 twelve-inch trees add up to exactly ten square feet of basal area per acre.

If the cruiser counts a second twelve-inch tree, the assumption is made that there are an additional 12.74 twelve-inch trees on every acre in the stand being sampled. The second tree counted is also worth ten square feet of basal area per acre. This second twelve-inch tree suggests the presence of another 12.74 twelve-inch trees on every acre. Now, 25.48 twelve-inch trees are assumed to be present on every acre in the stand, and together their basal areas add up to exactly twenty square feet per acre.

Thus far, we have restricted our example to twelve-inch-diameter trees. But the same relationship applies to trees of all sizes. Now consider a twenty-inch-diameter tree. The imaginary circle surrounding a twenty-inch tree also has a radius of thirty-three times the diameter of the tree—in this case, fifty-five feet. The cruiser can count a twenty-inch tree that is up to fifty-five feet away from the sampling point.

The imaginary circle of fifty-five-foot radius around the twenty-inch tree has an area of 9,503.25 square feet, which equals 0.2183 acre, or about 1/5 of an acre (more precisely, 1/4.58 acre). When the cruiser’s instrument picks up a twenty-inch tree, the assumption is made that there are 4.58 twenty-inch trees on every acre in the stand represented by the sampling point. Each twenty-inch tree has a basal area of 2.183 square feet. These 4.58 twenty-inch trees total up to exactly ten square feet of basal area per acre. Therefore, one twenty-inch tree counted by the cruiser is also worth ten square feet of basal area per acre. The three trees counted thus far represent thirty square feet of tree trunks per acre in the whole stand represented by the sampling point.

And so on, for every other diameter. The system assumes that each acre contains many small trees for each small tree counted, but only a few large trees for each large tree counted. The bigger the tree, the farther away from the sampling point it can be and still be counted. With a “10” basal-area-factor instrument, the cruiser could count a sixteen-foot-diameter giant sequoia a tenth of a mile away, but couldn’t count a sapling the diameter of a pencil from farther than about nine inches. That small tree would represent 23,256 pencil size saplings per acre, but only one sixteen-foot giant sequoia every twenty acres. It follows that, regardless of the size of the tree counted by the instrument, it always represents ten square feet of basal area per acre. (In practice, trees smaller than one-inch in diameter are usually ignored.)

The system works best when it is used to sample a relatively uniform stand of trees, in which trees of the same size occur again and again, evenly spaced throughout the stand. In reality, this isn’t usually the way it is in Pennsylvania hardwood stands. A real stand is likely to be more or less irregular, with trees unevenly spaced. To compensate for this variation, several sampling points are taken in a stand, and the totals from the different points are averaged. Note that the trees counted at each sampling point are totaled, but the totals at each point are not added together. They are averaged. Each sampling point is an estimate of the entire stand.

The point-sampling system will only estimate the density of stocking of a stand of trees on a per-acre basis. It assumes that each sampling point is representative of the whole stand, and gives the equivalent basal area per acre. It will not give the actual basal areas of the trees counted.

Prepared by Thomas J. Fitzgerald, Service Forester, PA DCNR Bureau of Forestry (Ret.)