A tree belongs to the property owner whose land the trunk is on. If it straddles two properties, it belongs to both owners. Any owner of the tree can trim branches on their property; this is also true even if the trunk is on your neighbour’s property, but the roots or branches extend onto your side (as long as you don’t harm the tree). Be aware that you have to stay on your own property to do this.
If a tree from your neighbour’s yard falls into your yard, who pays for the damage depends on the situation before the tree fell. If it was a healthy tree, you probably are stuck paying yourself as your neighbour would have had no way of knowing that the tree was at risk of falling. If the tree was dying or damaged, however, and if you had warned your neighbour to take it down, you may be able to recover some of the cost from your neighbour.