Posted: 05/07/2014 at 07:29
Speaking as someone who has a garden totally under-laid with roots from large mature trees, I can assure you that if the trees are alive the roots will regrow and you will be removing them every year! We dig a large number of small tree roots out of our veg patch every winter.
For the rest of the garden, we learn to garden with them. Where there are roots near the surface we have a 'wild garden' with ferns and native plants and flowers, under planted with spring bulbs - like a little patch of woodland.
The trick is to look at what you've got before you decide what sort of garden you want
Unless you're removing roots that require the use of a saw or an axe I don't think you'll harm the tree - unless you remove a great many of them - but they will regrow. Think of pruning a tree's branches - cutting them back just stimulates growth!
However, it would be polite, politic and just good neighbourly to explain the problem and mention what you're doing to the neighbours