how do I kill the roots of an ash tree as it's re seeding all over my lawn

I am not sure that I fully undferstand this question, Agatha, if you kill the roots you kill the tree! If the tree is a problem, shame, then why not just cut it down?

It would be a shame to get rid of what I assume is a healthy tree. Why not just treat the tree seedlings? I'm sure that they can't be too many and they're not prolific growers. A spot spray of systemic wed killer like Verdon now and again should sort the problem out.

Why would you want to kill the roots? Be thankful if the parent tree is healthy and dig up, pot on and share the seedlings.

If the seedlings are in the lawn then they can just be mowed off when cutting the grass.

Hmm, well this question isn't clear, but instinct tells me that either the tree itself has been chopped down, but the roots left in situ, or is dead, and the growth that's appearing in the lawn is suckering from the main roots, which are still alive. If that's the case, Agatha, I'm afraid you either need to have the roots removed, or you need to expose them, drill into them and fill the holes with SBK to kill them off.

I second that. Ash trees are a menace for seeding everywhere, even worse than sycamore if that's possible.