Sols, you've probably got a Yellow Pages: just look up Tree Surgeons, Garden Centres, etc. and ask them what the max is, and when the tree should be pruned to keep it to size, though there are so many varieties, they may not be able to tell you without knowing the species.
The trees rarely exceed 10m in height, but that can be kept down by having a tree surgeon cut back branches during the winter if he thinks it necessary. The roots won't be a problem unless the tree is growing right up to the neighbouring house, which it obviously isn't. Your neighbour sounds as if he wants the world to turn on his axis, so don't put up with any interfering from him.
Are you renting? If you are, tell your landlord about the man's remarks, and if the tree needs a trim, get the landlord to pay for it. Don't do it yourself without written permission from your landlord, anyway, as it's his tree. If the neighbour twitters on again, give him your landlord's phone number/address and tell him to take up his nitpicking direct.
If you own the property, the Council can send out a person from the Environment office to confirm that the tree is not infringing any boundaries, and that will be an end of it.