  Let $S_{p,n}$ denote the sample covariance matrix based on $n$ independent
identically distributed $p$-dimensional random vectors in the null-case. The
main result of this paper is an explicit expansion of trace moments and
power-trace covariances of $S_{p,n}$ simultaneously for both high- and
low-dimensional data. To this end we expand a well-known ansatz of describing
trace moments as weighted sums over routes or graphs. The novelty to our
approach is an inherent coloring of the examined graphs and a decomposition of
graphs into their tree-structure and their \textit{seed graphs}, which allows
for some elegant formulas explaining the effect of the tree structures on the
number of Euler-tours. The weighted sums over graphs become weighted sums over
the possible seed graphs, which in turn are much easier to analyze.
