I think you would get a good result using Illustrator's Extrude shape tool - this is found in file menu:
Effect > 3D
Make your type tread pattern the same length as the circumference of
the tyre shape you require - you then drag the tread shape in to the
Symbols panel. You need to do this because the Map Art function
finds resource assets here.
Make circle and extrude to a depth relative to a tyre shape (this can be tweaked later on with the Appearance dialogue box)
Set your Rotation axis and Perspective
You can then Map Art - this will bring open another dialogue box that allows you to select the facet on which you map your Symbol - your symbol will appear in a drop down menu - you will be able to manually rotate/stretch this in to position.
Once you are happy with the shape you can un-group and release the objects and select the line art you require.
A similar project can be viewed here available on Youtube
If you want your tread to sit on a two way curved, more torus-like surface (as the real tyre surface is) you can replace the 3d extrusion by 3d revolution. However, this might make the mapping more tricky.
Hope this helps - there are a fair few tutorials on this matter available on the web, Happy Mapping!