  Individual dark matter halos in cosmological simulations vary widely in their
detailed structural properties such as shape, rotation, substructure and degree
of internal relaxation. Recent non-parametric (principal component) analyses
suggest that a few principal components explain a large fraction of the scatter
in halo properties. The main principal component is closely linked with
concentration, which in turn is known to be related to the mass accretion
history of the halo. Here we examine more generally the connection between mass
accretion history and structural parameters. The space of mass accretion
histories has principal components of its own. We find that the strongest two
can be interpreted as the overall age of the halo and the acceleration or
deceleration of growth at late times. These two components only account for
$\sim70$%\ of the scatter in mass accretions histories however, due to the
stochastic effect of major mergers. Relating structural parameters to formation
history, we find that concentration correlates strongly with the early history
of the halo, while relaxation correlates with the late history. We examine the
inferences about formation history that can be drawn by splitting haloes into
subsamples, based on observable properties such as concentration and shape at
some final time. This approach suggests interesting possibilities, such as the
possibility of defining young and old samples of galaxy clusters in a rigorous,
quantitative way, or testing the dynamical assumptions of galaxy formation
models empirically.
