  We use a large $N$-body simulation to study the relation of the structural
properties of dark matter halos to their assembly history and environment. The
complexity of individual halo assembly histories can be well described by a
small number of principal components (PCs), which, compared to formation times,
provide a more complete description of halo assembly histories and have a
stronger correlation with halo structural properties. Using decision trees
built with the random ensemble method, we find that about $60\%$, $10\%$, and
$20\%$ of the variances in halo concentration, axis ratio, and spin,
respectively, can be explained by combining four dominating predictors: the
first PC of the assembly history, halo mass, and two environment parameters.
Halo concentration is dominated by halo assembly. The local environment is
found to be important for the axis ratio and spin but is degenerate with halo
assembly. The small percentages of the variance in the axis ratio and spin that
are explained by known assembly and environmental factors suggest that the
variance is produced by many nuanced factors and should be modeled as such. The
relations between halo intrinsic properties and environment are weak compared
to their variances, with the anisotropy of the local tidal field having the
strongest correlation with halo properties. Our method of dimension reduction
and regression can help simplify the characterization of the halo population
and clarify the degeneracy among halo properties.
