I'm here if you want to discuss your matter. I'll tell you what I think based on what you've said. If there's more information to add, then please reply.
You have the right to cut the branches from your neighbour's tree when they are hanging over your property line. However, there may be bylaws regarding it. I'm not aware of Ottawa having any special rules, unlike Toronto and London, but you should still contact the city's bylaw enforcement to ask.
So, if you are concerned that trimming the tree would damage the tree, you should discuss it with your neighbour. This is why I asked how you two get along. In order to make sure that the trimming you want to do won't damage the tree, you should have an arborist come and look at it and get an opinion. This shouldn't cost much, and with some diplomacy and luck your neighbour will agree that it's necessary and agree to split the cost. It's better to prevent damage and issues that can lead to lawsuits than to focus on what need in order to win the suit. By the time the neighbour sues you and you win, you'll have spend a lot of time and effort and perhaps money on legal fees, which is a waste and will ensure that you and your neighbour are forever at odds.
Even if the neighbour refuses to allow you to cut any branches and doesn't agree to split an arborist's costs, you should still call one to protect yourself. The arborist can likely give you an assessment just by looking at the tree from your property. Get the opinion in writing.
You're allowed to cut branches, as I wrote. However, if to do so causes the tree any serious or permanent damage, then the owner could sue you. As well, even if the tree survives the pruning (and that's why you want the arborist too, to tell you exactly how to do it so as to minimize damage to the tree) then your neighbour could still bring an action for reducing their property value due to the damage to the tree's aesthetic. That's rather nebulous, but if your neighbour is litigious this is certainly possible.
If you get the arborist's opinion in writing regarding whether pruning will damage the tree, and the best way to do the pruning, let your neighbour know. Give some written reasons for why you want to prune, and include the arborist's report. Suggest that you intend to prune, but if the entire tree was pruned back that the overall effect on the tree's aethetic would be reduced, rather than you simply pruning along the vertical plane of the property line, and if the neighbour agrees to get the whole tree done professionally then you'd share the cost.
These communications and offers aren't simply meant to keep you and your neighbour friendly. In the event that your efforts are rebuffed, and your prune, and your neighbour sues you, then you have good evidence that you went over and above reasonable efforts to minimize the damage to your neighbour while you did what you're allowed to do. It would go a very long way in court, I promise.
Now, if you want to prune the tree in order to install a windmill or solar panels or for some reason which you know that your neighbour won't like, then it's even more important that you take these steps because your neighbour is more likely to bring an action. You need to create your own evidence that you tried to be co-operative and to minimize damage, when when you were doing was lawful.
Your neighbour might say "forget it, I'm not paying for an arborist's report or for professional trimming", and that's fine. But if they say that, then they can't later reasonably claim a huge sum from you due to damage caused by your actions. If a large value was at stake then the neighbour should have spent some money to ensure its prevention, right? That will be your defense in court.
I hope that this answers you. If not then please reply. If so then I'd appreciate a positive service rating please. But I urge you to not simply leave your money on deposit with the site, because then it's gone.