Help! Tens of Tree Saplings all over lawn - how to kill?

Hi, I'm fairly new to gardening.  We bought our house 2 years ago and we have lots (in the tens) of tree saplings in our garden - mainly in the lawn.  We can't pull them up and the roots are usually too big for secaters so use loppers on them - even though they are only a couple of inches tall.  I just tried to pull a tiny one up and managed to but it also pulled up about a meter of lawn with a huge root system.  It obviously didn't pull all out so it made me think all the saplings were linked to the same root.  How do I kill them as I think they may be going under our conservatory and house.  I don't even know what tree they are, the leaves are soft and slighly serrated, green with a slight red tinge to the outer edge.

Any help and expertise would be greatly appreciated.  Thank you.

Can you post a photo? These sound like suckers from a tree rather than individual saplings, and you need to sort out where they are coming from. If you trim suckers, as you have been doing, they just grow more strongly than ever.

Hi Alina, after doing some research I think you may be right and they're suckers.  I've taken some photos but for some reason they're not that in focus, sorry.  Hope they help

I don't think it's one of the trees in our garden but there is a tree behind us, against the fence, which has a similar leaf.  I have pulled down a branch to take the photo below:

 Is there any way I can kill them and not kill the neighbours tree if they are coming from it?

not sure a way to kill them would be to buy some roundup stump treatment cut the top with secutares and paint some on and wiat this will not damage the lawn either if it doesnt touch it but this may effect the nieighbours tree

That's definitely a sucker. I'd be cautious about using a stump killer - that would probably affect the tree. Instead, expose the base of the shoot and cut it off as flat as you can to the root, or rip it off - that might be enough to stop it sprouting again.

My neighbour's plum tree sends up suckers all over my garden.  If they come up in the law I just mow them.  If they come up in a flower bed I try to cut out as much of that bit of root as possible.

I have had something similar, with hedging trees sending up suckers in my lawn. I have applied glyphosate (Roundup) to the leaves using rubber gloves and wiping it onto the leaves between finger and thumb, taking care not to get it on the grass. I leave the lawn uncut for a week afterwards. The glyphosate will gradually kill the part of the root system leading to the suckers, but won't kill the tree. It won't stop them coming back but will reduce them.

The only other help I can offer is to suggest putting a physical barrier along the edge of your property to stop the roots coming into your garden. I have had to do this to stop Symphoricarpus from the roadside. The roots will eventually get through anything, but it does give you some respite.

Thanks everyone, at least I know what they are and various ideas how to deal with them.