My toilet and bathroom sink are right next to each other and both are vented with a 2" pipe, with the separate run for the sink that connects about 4 feet up as I've seen in the diagrams for venting.

I can also run a vent for the bathtub but it will be a pain as it's across the room (about 5 ft).

Is it really necessary to have a vent for the tub or does the overflow drain let it drain well?

THANKS.