Front tyres worn flat across the centre of the tread can cause or contribute to head shake. I understand what happens is the tyre rides up on 1 edge of the flat section, then "falls" back to the low middle section and up onto the other edge. This phenomenon continues which is the head shake. Think of rolling the spare tyre for your car along in front of you. All is stable until it starts to weave slightly, then it just weaves further and further off the straight line. This is why new tyres can cure it.