\sqrt 2 Rule for Controlling the Tree Pattern in Forest Cut

A forest’s productivity can be optimized by the application of rules derived from monopolized circles. A monopolized circle is defined as a circle whose center is a tree and whose radius is half of the distance between the tree itself and its nearest neighbor. Three characteristics of monopolized circle are proved. (1) Monopolized circles do not overlay each other, the nearest relationship being tangent. (2) “Full uniform pattern” means that the grid of trees (a×b=N) covers the whole plot, so that the distance between each tree in a row is the same as the row spacing. The total monopolized circle area with a full uniform pattern is independent on the number of trees and $\frac{\pi }{4}$ times the plot area. (3) If a tree is removed, the area of some trees’ monopolized circle will increase without decreasing the monopolized circles of the other trees. According to the above three characteristics, “uniform index” is defined as the total area of monopolized circles in a plot divided by the total area of the monopolized circles, arranged in a uniform pattern in the same shaped plot. According to the definition of monopolized circle, the distribution of uniform index $(L) = \frac{{\chi ^2 (2n)}}{{2\pi n}}$ for a random pattern and $E(L) = \frac{1}{\pi }$ the variance of L is $D(L) = \frac{1}{{n\pi ^2 }}$ . It is evident that E(L) is independent on N and the plot area; hence, L is a relative index. L can be used to compare the uniformity among plots with different areas and the numbers of trees. In a random pattern, where L is equivalent to the tree density of the plot in which the number of trees is 1 and the area is π, the influence of tree number and plot area to L is eliminated. When n→∞, D(L)→0 and $L \to \frac{1}{\pi } = 0.318$ it indicates that the greater the number of tree is in the plots, the smaller the difference between the uniform indices will be. There are three types of patterns for describing tree distribution (aggregated, random, and uniform patterns). Since the distribution of L in the random pattern is accurately derived, L can be used to test the pattern types. The research on Moarshan showed that the whole plot has an aggregated pattern; the first, third, and sixth parts have an aggregated pattern; and the second, fourth, and fifth parts have a random pattern. None of the uniform indices is more than 0.318 (1/∏), which indicates that uniform patterns are rare in natural forests. The rules of uniform index can be applied to forest thinning. If you want to increase the value of uniform index, you must increase the total area of monopolized circles, which can be done by removing select trees. “Increasing area trees” are the removed trees and can increase the value of the uniform index. A tree is an increasing area tree if the distance between the tree and its second nearest neighbor is $\sqrt 2 $ times longer than that between the tree itself and its first nearest neighbor, which is called the $\sqrt 2 $ rule. It was very interesting to find that when six plots were randomly separated from the original plot, the proportion of increasing area trees in each plot was always about 0.5 without exception. In random pattern, the expected proportion of increasing area trees is about 0.35–0.44, which is different from the sampling value of 0.5. The reason is very difficult to explain, and further study is needed. Two criteria can be used to identify which trees should be removed to increase the uniform index during forest thinning. Those trees should be (1) trees whose monopolized circle areas are on the small side and (2) increasing area trees, which are found via the $\sqrt 2 $ rule.