This paper shows that one can use LP-based methods to minimize the width of a "balanced-layered" drawing of tree, but if one desires a grid drawing the problem becomes NP-hard. 

Lecture #3: reference for the NP-hardness proof regarding optimally "compactifying" an orthogonal drawing of an embedded graph. 

This paper contains several applications of the crossing lemma. Be warned that the presentation is a bit dense at times. 

Terry Tao's blog entry on the crossing inequality. 

This website gives a live demo environment (with configurable constants) and many example graphs drawn via force-directed methods.