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The results of Shekhar et al. (2017) "indicate that ASTRAL requires

$\mathcal{O}(f^{-2} \log{n})$ 

gene trees to reconstruct the species tree correctly with high probability where n is the number of species and f is the length of the shortest branch in the species tree.

How do I interpret this function, specifically the big O notation, when applying it to my data to estimate whether I have enough loci?

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    $\begingroup$ Please edit your question and provide a link or a full citation to the paper you are quoting. There are 86 papers in PubMed matching (Shekhar[Author]) AND ("2017"[Date - Publication] : "2017"[Date - Publication]) . Also clarify what you mean by applying this function. Why would you apply it? $\endgroup$
    – terdon
    Dec 7, 2018 at 11:29
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    $\begingroup$ I've updated your post and added a link to the actual paper you mean. A preprint is available here: arxiv.org/abs/1704.06831 $\endgroup$
    – Devon Ryan
    Dec 7, 2018 at 13:13

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Big O notation means "of the order of". Thus $\mathcal{O}(f^{-2} \log{n})$ means that the number of trees required goes with $\log{n}$ for number of species (i.e. if you square the number of genes, you should double the number of trees) and with the inverse square of the shortest branch length (i.e. if you double the shortest branch length, you quarter the number of trees needed).

Big O notation generally allows to to compare two input sizes, but cannot really tell you anything about a single point. I.e. it is hard (impossible?) to tell from this function whether you have sufficient loci.

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