P. Friedrich, in his article "On Aspect Theory and Homeric Aspect" in IJAL Memoirs 28, gives us many, many, many sentences like the one diagrammed in example (1). Now, while this diagrammed sentence is indeed not a reflection of the up-to-the-minute Generative Theory, it does have a resemblance to one of its previous incarnations, and will do for our purposes.
If we consider the semantics of this example sentence, which as we have noted, is representative of many others, we see that many of the words associated with differing categories are in fact all elements of a single category. Indeed, because this one category is so pervasive throughout this sentence, and constitutes the unified theme of the sentence, we hypothesize that there is a single node for this category, which is iconically labelled in the revised diagram in example (2).
From this meager in quantity but overwhelming in quality data, we must conclude that lines in tree diagrams must, in some circumstances, be allowed to cross.