Originally Posted by acesandeights
Typically, as the contact patch decreases the contact pressure increases. As you lose floatation you often increase bite, or traction due to the smaller contact patch. I have more experience in offroad truck tires, but if you want to increase traction, you often need a narrower tire. There are trade offs and conditions can dictate what works in a given situation, but riding desert sand might favor a wider tire, where you need more floatation, and forest single track might favor a narrower tire, where the narrower tire has more traction.
What is interesting about this is that, all other things being equal, which of course they almost never are, if the only change is the tyre width, keeping the same type and construction of tyre, and the same air pressure, as the tyre gets narrower, the pressure at the patch increases, which flattens the contact patch. The net result is that the surface area of the contact patch is more or less the same, but its shape gets progressively narrower and longer. Arguably more relevant when talking about 4 wheeler tyres which of course have a nominally flat tread.
A wider leading edge to the contact patch creates more flotation on soft surfaces. In fact sand is like going up a hill all the time. The tyre sinks in and is always climbing up the front edge. Make it narrower, and its a steeper/deeper face to climb out.
A longer contact patch can add stabilty compared to a wider one. Its why we prefer 21" fronts to smaller ones with wider tyres on our dirt bikes. The contact patch isn't much different in size, but the shape is longer, hence more directional stability.