Skip to main content
Log in

Invariants of phylogenies in a simple case with discrete states

  • Published:
Journal of Classification Aims and scope Submit manuscript


Under a simple model of transition between two states, we can work out the probabilities of different data outcomes in four species with any given phylogeny. For a given tree topology, if all characters are evolving under the same probabilistic model, there are two quadratic forms in the frequencies of outcomes that must be zero. It may be possible to test the null hypothesis that the tree is of a particular topology by testing whether these quadratic forms are zero. One of the tests is a test for independence in a simple 2×2 contingency table. If there are differences of evolutionary rate among characters, these quadratic forms will no longer necessarily be zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • CAVENDER, J.A. (1978), “Taxonomy with Confidence,”Mathematical Biosciences, 40, 271–280 (erratum,44, 308 (1979)).

    Article  Google Scholar 

  • CAVENDER, J.A. (1981), “Tests of Phylogenetic Hypotheses Under Generalized Models,”Mathematical Biosciences, 54, 217–229.

    Article  Google Scholar 

  • DAYHOFF, M.O., and ECK, R.V. (1968),Atlas of Protein Sequence and Structure 1967–1968, Silver Spring, Maryland: National Biomedical Research Foundation.

    Google Scholar 

  • FELLER, W. (1968),An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., New York: Wiley.

    Google Scholar 

  • FELSENSTEIN, J. (1981), “Evolutionary Trees from DNA Sequences: A Maximum Likelihood Approach,”Journal of Molecular Evolution, 17, 368–376.

    Article  Google Scholar 

  • FELSENSTEIN, J. (1983a), “Inferring Evolutionary Trees from DNA Sequences,” inStatistical Analysis of DNA Sequence Data, Ed. B.S. Weir, New York: Marcel Dekker, 133–150.

    Google Scholar 

  • FELSENSTEIN, J. (1983b), “Statistical Inference of Phylogenies,”Journal of the Royal Statistical Society, A146, 246–272.

    Google Scholar 

  • FELSENSTEIN, J. (1985a), “Confidence Limits on Phylogenies with a Molecular Clock,”Systematic Zoology, 34, 152–161.

    Google Scholar 

  • FELSENSTEIN, J. (1985b), “confidence Limits on Phylogenies: An Approach Using the Bootstrap,”Evolution, 39, 783–791.

    Google Scholar 

  • NEYMAN, J. (1971), “Molecular Studies of Evolution: A Source of Novel Statistical Problems,” inStatistical Decision Theory and Related Topics, Eds. S.S. Gupta and J. Yackel, New York: Academic Press, 1–27.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

The authors wish to thank Thomas Haigh and an anonymous reviewer for helpful comments on the manuscript. Work by J. Felsenstein was supported in part by task agreement number DE-AT06-76EV71005 of contract number DE-AM06-76RL02225 between the U.S. Department of Energy and the University of Washington.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cavender, J.A., Felsenstein, J. Invariants of phylogenies in a simple case with discrete states. Journal of Classification 4, 57–71 (1987).

Download citation

  • Issue Date:

  • DOI: