  We make a detailed study of various (quadratic and linear) Morse-Bott trace
functions on the orthogonal groups $O(n)$. We describe the critical loci of the
quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect
fillings of tables associated with the multiplicities of the eigenvalues of $A$
and $B$. We give a simplified treatment of T. Frankel's analysis of the linear
trace function on $SO(n)$, as well as a combinatorial explanation of the
relationship between the mod $2$ Betti numbers of $SO(n)$ and those of the
Grassmannians $\mathbb{G}(2k,n)$ obtained from this analysis. We review the
basic notions of Morse-Bott cohomology in a simple case where the set of
critical points has two connected components. We then use these results to give
a new Morse-theoretic computation of the mod $2$ Betti numbers of $SO(n)$.
