[snip]

I *think* that you are permitted to remove all parts of the tree overhanging
your boundary but this is mitigated by, for example, protection orders.

Of course you would be well advised to seek legal advice before embarking on
any such venture.

A better tack, depending on where you live, may be to point out that if/when
these trees cause subsidence to your house, the council will be responsible.
Where I live (London clay all around) the council keep most trees near
houses well under control. It may help but if you have a dense row of them,
and they were there when you moved in, you probably have little you can do.

How about growing shade loving plants and giving up on the grass in favour
of a deep bark play area? The kids won't care much and it's far easier to
work with.

Paul DS.