A seven-branched sword can be used to snag opponents’ clothing or armor.
To snag armor or clothing, the attacker makes a trip attempt.
If successful, the victim doesn’t fall prone, but instead is snagged and stumbles forward, leaving the victim flat-footed for the remainder of the round.
The question is: can you use a seven-branched sword in this way against opponents that cannot be tripped, like a flying foe, or a creature with no legs, etc.?
Unfortunately, no. Not without GM fiat.
It would be a perfectly reasonable houserule to allow a generic combat maneuver check, rather than the called out Trip maneuver. Think of Stand Still, or the Tengu Swordmaster Tiger Trance where you simply make a CMB vs CMD check.