Recent work on high dimensional graphical modeling of continuous data
has been focused on Gaussian models with sparsity constraints imposed
on the inverse covariance matrix.  In this work we relax the Gaussian
assumption, but restrict the underlying undirected graph to be a
forest.  This allows fully nonparametric density estimates, but with
structural assumptions on the graph.  

Since fitting a fully connected spanning tree can be expected to
overfit in high dimensions, we regulate the complexity of the model by
cross validating over a large class of spanning forests of different
sizes. One class of forests comes from pruning the maximum weight
spanning tree.  We also consider spanning forests with restricted tree
sizes.  We prove that finding a maximum weight spanning forest with
restricted tree size is NP-hard, and develop an approximation
algorithm.  We use the algorithm to build a second class of forests by
pruning the approximately optimal tree-restricted forest. Finally, we
also compare with the pure greedy approach.

We analyze the resulting density estimation scheme in the high
dimensional setting, allowing both the sample size and dimension to
increase, and prove oracle results on the risk and structure selection
properties of the method.