Appendix S1. Conditions for the concavity of Q.

Appendix S2. Analytical solution for calculating (i) the expected value of the Rao quadratic diversity \overline Q _{(M) and (ii) the intersection between distinct rarefaction curves.

Appendix S3. Proof that, for any concave diversity index Q, for a complete enumeration of all possible combinations of M plots out of N (NM), a non-decreasing accumulation curve of the expected diversity \overline Q _{(M) vs. M is obtained.

Appendix S4. R function ‘Rare_Rao’ for the calculation of functional rarefaction.

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