I never thought that the day would come when I would verify an uncertainty posed by the redoubtable John_T, but it is not a tale. It is absolutely true that the U.S. Army adopted a standard tree height of 50 ft, and instituted a takeoff to 50 ft requirement as a result.
So too, 411A has correctly provided the origins of the latter day 35 ft requirement.
To add one more, the 35 ft requirement reverts back to the standard 50 ft if the takeoff involves a curved procedure. This is to account for the wing tip being lower than the usual lower point of the aircrfat in a banked turn. With the advent of the A380, this too, may change.
Regards,
Old Smokey