Yes, although it is best to negotiate with your neighbour prior to taking any action. You can cut any branches or roots that cross over the boundary. However the branches/roots still technically belong to your neighbour, so you should speak to them to see if they want them prior to disposing of them.
If you want to cut any parts off a tree then you should check prior to doing this, that the tree is not subject to a tree preservation order. You need to get consent from the local authority before pruning such a tree.
Boundary issues can be a very emotive issue and in all cases, discussion with your neighbour is always advised.