Tree Shoots with long Roots In Our New Garden

Hiya.
We have moved into a flat with a garden and there are tree shoots all over the place, some of them were almost as tall as me.
We have tried pulling them out but there are some bloody long roots- some going under our patio - so I just trim them down and try and keep trimming everytime they sprout up again.
Because the previous occupant struggled for 8 years to get the grass to grow (large patches and they even laid fresh turf and it died in the same patches) we have thought about putting artificial turf down, but are worried about these little tree shoots piercing it and coming back.
Will this happen?
Is there anything we can do to stop them returning?
The main culprits are from two trees in two adjoining gardens.
We have moved into a flat with a garden and there are tree shoots all over the place, some of them were almost as tall as me.
We have tried pulling them out but there are some bloody long roots- some going under our patio - so I just trim them down and try and keep trimming everytime they sprout up again.
Because the previous occupant struggled for 8 years to get the grass to grow (large patches and they even laid fresh turf and it died in the same patches) we have thought about putting artificial turf down, but are worried about these little tree shoots piercing it and coming back.
Will this happen?
Is there anything we can do to stop them returning?
The main culprits are from two trees in two adjoining gardens.
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These photos are very dark as it is about to rain again, but they appear to be from this tree. Hopefully you can tell what tree it is still.
Might be a form of Robinia, hard to say, but it does sucker.
That was the only one I knew with a very deep root that you can’t pull out.
Theres one just up the road from here ... it’s vast!
Gardening in Central Norfolk on improved gritty moraine over chalk ... free-draining.