  The observation that a graph of rank $n$ can be assembled from graphs of
smaller rank $k$ with $s$ leaves by pairing the leaves together leads to a
process for assembling homology classes for $Out(F_n)$ and $Aut(F_n)$ from
classes for groups $\Gamma_{k,s}$, where the $\Gamma_{k,s}$ generalize
$Out(F_k)=\Gamma_{k,0}$ and $Aut(F_k)=\Gamma_{k,1}$. The symmetric group
$\Sigma_s$ acts on $H_*(\Gamma_{k,s})$ by permuting leaves, and for trivial
rational coefficients we compute the $\Sigma_s$-module structure on
$H_*(\Gamma_{k,s})$ completely for $k \leq 2$. Assembling these classes then
produces all the known nontrivial rational homology classes for $Aut(F_n)$ and
$Out(F_n)$ with the possible exception of classes for $n=7$ recently discovered
by L. Bartholdi. It also produces an enormous number of candidates for other
nontrivial classes, some old and some new, but we limit the number of these
which can be nontrivial using the representation theory of symmetric groups. We
gain new insight into some of the most promising candidates by finding small
subgroups of $Aut(F_n)$ and $Out(F_n)$ which support them and by finding
geometric representations for the candidate classes as maps of closed manifolds
into the moduli space of graphs. Finally, our results have implications for the
homology of the Lie algebra of symplectic derivations.
