A TRE is an ignition "timing retard eliminator" - the stock ignition maps on most Suzuki's include built in ignition retard to help control wheelies, wheelspin and power in 1-4th gears, and top speed in 6th gear. 5th gear is the only "open" gear. The TRE is a resistor that fools the ECU into thinking it is in 5th gear all the time. That is the basic concept. From there, folks have made some other modifications to expand on the abilities of the modules.... Smart TRE's, X-TRE, etc...
I would say this is a street legend that is being repeated. The fact is that the timing is affected only with <10% throttle openings so must be a heck of a bike to make wheelies with only a cruising throttle opening.
From my opinion the timing is altered for cruising area to control the emissions, make exhaust gasses burn better at the area where also pair is on and excess fuel is put throught the engine. The alledged performance gains may relate to keeping the fuelling same for all gears in the dyno - in real life with ram air compensation I would be rather suspicious about any real gains, rather the opposite - particularly as it has been documented that some riders have experienced worse ET:s on track after installing a TRE.
Another fact is that by fooling the ECU to consider the bike being on 5th gear you also let the engine rev higher when 6th gear is on. But as the peak power with stock and moderately tuned engines is often coming at 9000rpm it would be more beneficial to have a properly geared than TREd bike for LSR top speed runs. (For a quartermile something is horribly wrong if you hit 6th gear limiter before the trap.)
Sorry for my biased and a bit black and white opinion on this (based on facts and personal experience), I would say that better acronym would be:
The
Retarded
Enginesystem when talking about a TRE.
The only TRE I would ever consider is something that allows my engine to run N,1,2,3,4,5,5 - but because of having unrestricted many gen1 ECUs and having an ability to unrestrict gen2 ECU just dont need one anyway and therefore dare to say my opinion. I hope being this straight forward does not offend anyone.