What are the implications for training a Tree Ensemble with highly biased datasets? I have a highly biased binary dataset - I have 1000x more examples of the negative class than the positive class. I would like to train a Tree Ensemble (like Extra Random Trees or a Random Forest) on this data but it's difficult to create training datasets that contain enough examples of the positive class.

What would be the implications of doing a stratified sampling approach to normalize the number of positive and negative examples? In other words, is it a bad idea to, for instance, artificially inflate (by resampling) the number of positive class examples in the training set?
I would recommend training on more balanced subsets of your data. Training random forest on sets of randomly selected positive example with a similar number of negative samples. In particular if the discriminative features exhibit a lot of variance this will be fairly effective and avoid over-fitting. However in stratification it is important to find balance as over-fitting can become a problem regardless. I would suggest seeing how the model does with the whole data set then progressively increasing the ratio of positive to negative samples approaching an even ratio, and selecting for the one that maximizes your performance metric on some representative hold out data. 

This paper seems fairly relevant http://statistics.berkeley.edu/sites/default/files/tech-reports/666.pdf it talks about a weighted Random Forest which more heavily penalizes misclassification of the minority class.