laying new lawn

[KAlanc]

I have a lawn which contains a few scruffy patches. The garden is also uneven in some places so I am considering laying a new lawn after levelling the area. However, upon digging up a small section I discovered a whole web of roots quite close to the surface which I suspect belong to a mature ash tree at the rear of the garden. I also think this small section is representative of the whole garden. Could the tree be the reason for the "patchiness" or could a lawn still be properly maintained despite the proximity of such a large tree? I wouldn't want to go through the expense of a new lawn if after some time the same patchiness will occur.