"Nearest-tree" estimations - A discussion of their geometry

The use of “nearest-neighbor” sampling has a long history. It involves measuring the distance from a random point in an area to the nearest object. That history involves never quite solving the problem, many examinations of special cases that never occur, adjustments that were ad-hoc, and a great deal of uninformative algebra. In forestry we have attempted to use the “nearest-tree” method for estimating numbers of trees on a landscape but the method is general, and can be used for any objects being sampled. I believe that the literature has never shown the logic and geometry in a form that is useful to both understand and solve the problem. This paper discusses the method from the geometric point of view, making no assumptions about tree distribution, and shows why extending the processes to the “n closest tree” much reduces the bias and variability, as well as specifying what is needed to solve the problem in an unbiased way.