I would recommend to use diversity profiles of both control and treatment to compare the different aspects of diversity. There is a recent consensus favouring the use of diversity profiles for the measuring and comparison of diversity. The diversity profile combines different known indexes or measures of diversity into a family of diversity measures, as a function of a parameter that controls the sensitivity of a measure to species relative abundance. You can find more information in Chao et al 2012 (http://dx.doi.org/10.1890/11-1817.1) and references therein.

There is a great example in:

Jost, L., Chao, A. & Chazdon, R.L., 2010. Compositional Similarity and Beta Diversity. In A. E. Magurran & B. J. McGill, eds. Biological Diversity Frontiers in Measurement and Assessment. Oxford, UK: Oxford University Press, p. 368. (if interested, let me know I can email it to you)

Also, Gotelli and Chao’s (2013) chapter in the Encyclopedia of Biodiversity explain it very well.

Greetings

Have you ever considered such an index by Bray-Curtis?

I have also found Bray-Curtis to be a useful index.  It accounts for abundance of different species whereas Jaccard only accounts for presence/absence. For Sorensen, I am not sure, but one version I am familiar with also only accounts for presence/absence. The book "Measuring Biological Diversity" by Anne Magurren is a good reference to look into these types of measures.

Thanks for your expert answers.....There is a huge number of indices used for biodiversity comparison at different sites....But the question is If you want to compare biodiversity richness and composition at invaded and non invaded plots (following quadrat method) at the same site (geographical location), means data collection in invaded plot and then in non invaded plot in vicinity (as control) of first plot, what indices would you prefer for quadrat level biodiversity comparison to measure effect of particular plant species invasion for estimation of phytodiversity loss due to that particular invasive.....I got different papers following the same methodology but different biodiversity indices. Furthermore, are these indices sample size dependent? 

The two links below may possibly help you get some clarity Huma

http://www.planta.cn/forum/files_planta/chapter_8_analysis_of_differences_in_species_composition_172.pdf

http://www.econ.upf.edu/~michael/stanford/maeb5.pdf

This link may possibly be of some help to your questions

http://www.academia.edu/2239774/Plant_species_diversity_in_alien_black_locust_stands_A_paired_comparison_with_native_stands_across_a_north-Mediterranean_range_expansion

In agreement with Christopher's answer... the two measures you mentioned (Jaccard and Sorensen) just account for the 'asociation', i.e. the similarity based in the shared presences. These would give some idea on the relatedness of communities, but without incorporating anything on the species abundances which would be of interest to characterise communities...

All depends on the kind of data you have (binary, semiquantitative or counts). I also agree that the book by Magurran is a good source for ideas. I have seen one version in the web. The 'structure' could be measured using a diversity index like Shannon (H'). The composition would be a different think; if you had counts you could use Euclidean distances or a comparable measure...

All indices are sample size dependent to some extent, but the degree of sample size dependence varies depending on the particular data. For example, in one study I found that using Bray-Curtis, the average of many small comparisons gave about the same value as one large comparison using all the data. In a second study, this wasn't true.

However, you shouldn't get hung up on size dependence.  Instead, just make sure to compare sample units of the same size.  If you compare multiple units (quadrats) and give standard deviation, the reader will have a good idea of what the data looks like.  Cao et al. (1997) concluded that "independence of sample size is not a desirable attribute...real similarity between replicate sample sets increases with increasing sample size."  In your study what is important is probably not the absolute number, but comparing: non-invaded to non-invaded plots vs. invaded to non-invaded plots. 

Cao et al. (1997) provides interesting analysis of sample size dependence, and strengths and weaknesses of some indices. For example, they concluded that Morisita index gave far too much weight to very abundant species.

http://www.ingentaconnect.com/content/wef/wer/1997/00000069/00000001/art00013

Cao, Y, P Williams & AW Bark 1997 "Effects of sample size (replicate number) on similarity measures in river benthic Aufwuchs community analysis" Water Environmental Research 69: 107-114. 

Check out my "Statistical Ecology...", free download on my ResearchGate page.

I would recommend to use diversity profiles of both control and treatment to compare the different aspects of diversity. There is a recent consensus favouring the use of diversity profiles for the measuring and comparison of diversity. The diversity profile combines different known indexes or measures of diversity into a family of diversity measures, as a function of a parameter that controls the sensitivity of a measure to species relative abundance. You can find more information in Chao et al 2012 (http://dx.doi.org/10.1890/11-1817.1) and references therein.

There is a great example in:

Jost, L., Chao, A. & Chazdon, R.L., 2010. Compositional Similarity and Beta Diversity. In A. E. Magurran & B. J. McGill, eds. Biological Diversity Frontiers in Measurement and Assessment. Oxford, UK: Oxford University Press, p. 368. (if interested, let me know I can email it to you)

Also, Gotelli and Chao’s (2013) chapter in the Encyclopedia of Biodiversity explain it very well.

Greetings

I would use Bray-Curtis similarity index for comparison of species composition between sites. For comparing structure one can use Importance Value Index and sketch of profile diagrams.