Depending on the soil (sand/hard pack clay , etc) in the deserts here in Idaho most guys run the Dunlop Geomax AT81
This is what Dunlop says about its usage , it comes in 18"/19" and 21"
Details
Dunlop's vast championship experience and development expertise has been infused into the next generation of off-road tires: the all-new Dunlop Geomax® AT81™. These new Geomax tires offer more all-around performance plus higher levels of ruggedness for off-road racing and riding.
AT81 rear tread block shape and distribution plus DIRTuitive Grip Design (DGD) knobs help the tire penetrate down through the surface dirt for extra traction across a broad range of off-road applications.
Formulated high-wear-resistant rear tire compound offers enhanced chipping, tearing and wear appearance.
Geomax AT81 rear tire carcass features two nylon plies.
Front tire construction includes two plies of nylon wrapped with a third ply of polyester for a strong base.
Lateral grooves on the shoulder knobs provide additional biting edges and also allow the knobs to be more flexible.
Rear center blocks are arranged in-line to provide a larger contact patch for maximum traction.
Dunlop's "plush pad" recess design between tread blocks enhances compliance for bump absorption and ride comfort.
As far as the tire size goes , it depends on the brand and style of tire , some 110's are wider than others and some 120's are really wide and cause clearance issues especially when low air pressure is used , a 110 will dig in but tends to get to the edge quickly when leaning (its a narrow quick lean because its a round carcass) , a 120 will give you more surface area to stay on top of the dirt better (like in sand) but tends to rub the side knobs on the chain sometimes when flexing and when it gets to the edge it tends to get sketchy as it leans till it flexes (its flatter so when it gets to the edge its a abrupt transition and traction goes away (look at this mouth on the smiley face , see how its a wide curve then it almost goes back on itself at each edge ?

) that is how a 120 is , , it has a wide bite range till you lean just a little too far , then nothing , then its sliding and your on the ground , basically you have lots of bit , till you don't then its too late
I personally run 110/90 on my 450R , and i run a Kenda Equilibrium (trials/knobby hybrid) on my X and it does well everywhere , except it does not like abusive throttle , as it will wear quickly if you abuse it heavily , front i run 80/100 on both bikes , which i ride in both desert and single track , my R i run on hard pack MX tracks as well as hard pack and sand desert and single track , and use a Pirelli MX32 Pro and a MX Xtra X on the front and it works well in all those conditions , but the MX32 Pro only comes in 19"
Hope that helped ....
