Roots or Branches under lawn???

I have joined this Forum as I am not a very keen gardener but really need help.
For the past few weeks I have noticed shoots coming up through my lawn. They looked like they might be small shrubs.
Yesterday I thought I would dig them out and was horrified to find that all the shoots are coming from what appear to be roots or branches a few inches underground. I do have apple trees on that side of the garden but the roots/branches are coming under the fence, emanating from my next door neighbour's garden, under my narrow flower bed and then under the lawn. We have a six feet high fence and we don't talk a lot but are not on bad terms. They have a neat, well kept garden, so it is not overgrown
These "roots" are about 2 inches in diameter and get thinner as they spread under the lawn. I pulled one lot up and it has wrecked my lawn as it was the size of a good sized tree branch.
The thinner parts of it just broke off.
There are still more of the shoots so there must be more of the roots or branches under the lawn.
I say "branches" because there are no tendrils and they are not white, but have a covering of bark. Once I dug them up and they dried out they really look like branches.
Somebody suggested it might be Ground Elder, but the leaves are totally different, thankfully.
There is only one tree in my neighbour's garden, but that is some kind of ornamental fruit tree and again, it's spade shaped leaves are totally different.
Could it be a root stock of the neighbours (grafted?) tree sending out branches underground?
Has anybody any ideas? what is it? How do I get rid of it? Will it be difficult?
Pictures attached.
Picture of dug up root (or branch). Length was about 15 feet. Leaves top left are from another plant
Picture of shoot on dug up stem
Picture of my apple tree and neighbour's trees leaves.

 

Welcome to the forum Puzzled. You've got a very tricky problem there and I'm not sure what to suggest to you.
I think what is happening there is you have some tree roots very close to the surface and they have probably been nicked with the lawn mower and suckered - hence the shoots coming up all over your lawn.
Cherry trees are very guilty of that kind of behaviour - is that what your neighbor has. I can't make it out from the pic.
The leaves on the bit of root you dug up are definitely not apple so not from your tree.
I think all you can do is dig it all up and trace it back to its source at the boundary and cut the root off. But therein lies a problem. You can cut the roots which are in your property but your not allowed to damage the tree and it might be weakened and unstable with the roots cut off. And shame about your lawn.
See what others on here think. Somebody might have a good idea.
Once you have some kind of plan I would suggest a friendly chat with your neighbor and see if you can come to some kind of mutual agreement that includes the work and expense.
I hope it works out well for you.

 

They are cherry. I recognise the shoots on the roots. I had that after I planted cherries but not on the lawn. The leaves look quite different to those of the parent plant. I have a feeling your neighbours are not going to want to take their tree down. Other members would advise I am sure but do they not just mow off? If you are mowing the lawn regularly the shoots are being deprived and will just disappear. Cherries are not really suckering trees by and large. Are the roots down deep or close to the surface?