I've heard of people tieing an empty rubber wheel/tire to either side of the saddle, and letting the horse teach itself not to run into a tree. But I don't think that's the kind of advice you were looking for
A lot of neck reined horses I know are really good at working off leg and seat, and the reins are just an 'enforcer', of sorts. If they're not really good at neck reining, or working entirely off your other cues, I have never had a problem with picking up the reins (or a rein) when really needed, even though you're not technically supposed to.
Is it someone else's horse you're riding? I ask because you said you're not familiar with neck reining. If so, perhaps ask the owner to show you the 'buttons'.
If it were me, assuming it was a well-trained horse, and the tree were coming up on my left, I'd put my left leg on or just slightly behind the girth, asking the horse to sort of go sideways or leg yield a bit. To keep it from just moving directly off (since I asked for more of a lateral movement), I'd use rein against the neck to keep the horse's body straight - if needed.