## 2 Materials and Methods
### Methods
#### 2.7.3 Measures of Biodiversity  
Biodiversity is the quantitative measure of the variety of distinct species in an area and their relative abundance, a biological phenomenon described with consideration of both spatial and temporal scales [4]. Alpha diversity (\(\alpha\)-diversity) was developed by [PERSON] during his study of plant communities in the Klamath region to describe habitat (local) level diversity characteristics. It was introduced with the complementary statistics, \(\beta\)-diversity and \(\gamma\)-diversity. Beta diversity is dimensionless, as a comparative statistic, i.e., the ratio between \(\gamma\) (regional) and \(\alpha\) (local) diversities [48]. \(\alpha\)-diversity and \(\gamma\)-diversity are limited to discrete units of space--roughly communities and ecosystems, respectively [4], but the ratio describes the number of unique habitats utilized by species in a region [49].  
[PERSON] postulated that total species diversity in a landscape, described by \(\gamma\)-diversity, is determined by the mean species diversity at a community scale (\(\alpha\)-diversity) and also by the differentiation among those communities (\(\beta\)-diversity) [48; 50]. [PERSON]'s subsequent usage of \(\alpha\)-diversity implies the application of the statistic across multiple sites in a landscape, strongly influencing its primary use at the assemblage and community scale [48; 50].  
Insofar as the describing an observation of diversity present in a discrete unit of space, three broad types of biodiversity measures in the literature indicate an evolution in the understanding of biodiversity and related ecological processes. These are taxa based (taxonomic), trait based (functional, phenotypical expression of traits) and phylogenetic diversity, the latter defined as the minimum total length of phylogenetic branches required to span a given set of taxa on the phylogenetic tree [51].  
We used taxonomic diversity indexes for relative ease of calculation and the extensive theoretical background in the literature, but the others are mentioned in the discussion as they represent improvements on the usefulness of a spatial model. That said, a fundamental ecological axiom is that habitats differ in the richness and abundance of species associated with them. Therefore, some issues arise in the most basic taxonomic diversity calculations which must be addressed. These are (1) that the total number of species in a sample varies both within habitats and between different habitats and that unless adjustments are made it is difficult to compare taxonomic \(\alpha\)-diversity values, and (2) species of lesser abundance should not carry as much weight as those with higher abundance as their functional role is less [49].  
Point samples were acquired within a uniform mixed-conifer habitat type, and Menthinck's Index was used to provide an adjustment to species richness data. It measures point diversity as the ratio of number of species (s) and the square root of the total number of individuals (\(N\)). It is notated:  
\[MDI=\frac{s}{\sqrt{N}} \tag{13}\]  
where:  
* \(s\) is the number of different species in your sample;
* \(N\) is the total number of individual organisms in the sample.  
To assess the impact of species abundances, the Simpson Index relates the contribution made by each species to the total number of individuals present. This is notated:  
\[SDI=1-\sum_{i=1}^{s}p_{i}^{2} \tag{14}\]where \(p_{i}\) is the proportion of individuals found in species \(i\).  
_SDI_ is a non-parametric index value less sensitive to species richness in that species specific abundances are considered. As _SDI_'s summation increases, evenness decreases. The difference between that value and 1 produces a range of 0-1, 1 being a monoculture.