I was watching a show on the Wings channel the other night, and they showed some rich entusiast having a copy of a WWII plane built from scratch - only in three quarters scale. There was another show on recently about someone else who built a replica of a WWI German fighter from scratch - also in 3/4 scale. If you are going to spend the money, why bother making the duplicate 1/4 smaller - it surely can’t be that much less expensive?
I can think of three reasons:
- Some people are just crazy about models, they like the real thing, but prefer a model.
- There´s no need for a FAA certification to fly the plane. As a concequence the model doesn´t need to comply to some safety specifications that involve further expense, complexity and construction time.
- A 3/4 model probably weights half as much or less as the real plane, thus needs much less power, that means a much smaller engine that is much cheaper to operate, and so forth…
Or they are just crazy. 
Maybe this link to a relevant discussion on newsgroups (at Google) will work. The most interesting reason:
Regarding my second point I think I should clear that. What I meant to say about an FAA certification is that a reduced model would qualify as an experimental airplane, so it doens´t need certification. On the other hand (my WAG) an exact replica should be withheld to the same requirements and certifications as the original plane.
A scale replica and a full-sized replica would both need FAA airworthiness certification, most likely as experimental amateur or experimental exhibition. Ale’s point #3 is probably most accurate, judging from the homebuilders I know. A full-sized P-51 replica is a lot of plane - expensive and time-consuming to build, certify, insure and operate. A 3/4 or 2/3 scale P-51 (and there are a lot of plans and kits for them out there, using all sorts of materials) can be cheaper, perform about the same as a small production general aviation plane (so easier for normal private pilots to handle), and do it behind a smaller, quieter and more efficient engine. Some of these kit companies make an array of scale ‘warbirds’ with common parts and methods; for an example try doing a search for Loehle. I’m on the road and don’t have my copy of Aerocrafter handy, but I’ve seen P-51 replicas in tube and fabric, aluminum, wood, and carbon-fiber composite. Obviously the major common theme is the appearance of a P-51. Even the smaller WWI types often get scaled down. I have also seen a number of full-scale replica kits including a couple J-3 Cubs and the DeHavilland Chipmunk.
Very seldom are exact scale planes tried. Experience has shown that they do not scale down well. Most who build scaled down War Birds do so from a $$$ standpoint as was noted.
They also usually have a much different airfoil for safety, handling, and performance issues need for the smaller power plants.
Even a ½ scale P-51 would require an 7-800 HP engine to be ‘scale’. Big $$$$$$$ to attempt to produce that.
They are fun and if you want something that just everybody does not have…
Experimental does not equate with being outside FAA controls and inspections. Just that you do not have to prove it is a safe design, just that it is built with proper methods. There are a lot of restrictions on experimental aircraft.