  We provide a complete description of possible covariance matrices consistent
with a Gaussian latent tree model for any tree. We then present techniques for
utilising these constraints to assess whether observed data is compatible with
that Gaussian latent tree model. Our method does not require us first to fit
such a tree. We demonstrate the usefulness of the inverse-Wishart distribution
for performing preliminary assessments of tree-compatibility using
semialgebraic constraints. Using results from Drton et al. (2008) we then
provide the appropriate moments required for test statistics for assessing
adherence to these equality constraints. These are shown to be effective even
for small sample sizes and can be easily adjusted to test either the entire
model or only certain macrostructures hypothesized within the tree. We
illustrate our exploratory tetrad analysis using a linguistic application and
our confirmatory tetrad analysis using a biological application.
