What I propose to use as the parser representation instead of trees is a collection of constructions. I call it a cloud.
Base constructions are words. When several constructions (e.g. words) tend to go together and comprise something meaningful in the semantics, they form a composite construction and become its arguments, or children. Composite constructions may also be children of even higher-order constructions, although not very often.
Consider I see a little silhouetto of a man as an example. Each word is a construction, so it's already in the cloud. I+see form a construction whose meaning is to specify see's subject as I. Similar things happen with see+silhouetto, though this time it's an object. Both a and little provide us with additional information about silhouetto: it's indeterminate and, well, little. Again, indeterminacy is added to man in a+man construction. The preposition of+man form a prepositional construction, which then itself forms a genitive construction with silhouetto. The result can be nicely visualized:
Isn't this looking like a strangely-colored cloud?
Of course, the composite constructions don't need to have exactly 2 arguments. For example, the more X the more Y construction has 6 arguments, 4 of which are constant words. X is Y to Z (as in John is hard to convince) has 5 arguments with 3 slots (actually 4: is also isn't fixed, it's replaceable with was).
And yes, clouds are better than trees.