  We introduce a mean-field model of lattice trees based on embeddings into
$\Z^d$ of abstract trees having a critical Poisson offspring distribution. This
model provides a combinatorial interpretation for the self-consistent
mean-field model introduced previously by Derbez and Slade, and provides an
alternate approach to work of Aldous. The scaling limit of the mean-field model
is integrated super-Brownian excursion (ISE), in all dimensions. We also
introduce a model of weakly self-avoiding lattice trees, in which an embedded
tree receives a penalty $e^{-\beta}$ for each self-intersection. The weakly
self-avoiding lattice trees provide a natural interpolation between the
mean-field model ($\beta=0$), and the usual model of strictly self-avoiding
lattice trees ($\beta=\infty$) which associates the uniform measure to the set
of lattice trees of the same size.
