I'm Lucy, and I'd be happy to answer your questions today.
If the TRUNK of the tree straddles the boundary line, then the tree is a boundary tree and cannot be removed without the consent of both owners of the property. If they tried to take it out, you could sue for the costs of replacement. However, if it's minor roots or branches overrunning the property line, those CAN be removed by the neighbor. They're allowed to trim the tree back to your property line, because the tree is technically trespassing. Look very carefully at the base of the trees and shrubs in question, and the property line, and take pictures if necessary so you can show that these are shared trees/shrubs.
If the base of the trees/shrubs isn't on both properties, the way to save the tree would be more explaining about how the trees help with erosion and runoff to both properties, and how they benefit the land. That may require talking to a landscaper who can give an expert opinion on how having the trees/shrubs there benefits BOTH properties. In that case, you're really looking to persuade them to do the right thing and let the trees stay - but they can't make you remove entire trees/shrubs unless they're dead (because dead trees present a danger to the neighboring properties).
It is possible to argue that the land beneath the trees belongs to you because they remained on the property, openly and notoriously, without permission from the neighboring owner, for more than 7 years. However, in that scenario, you'd have to prove that the previous owner objected to the trees/shrubs. Also, the new owner would be able to sue you for a rebate on property taxes for every year the trees/shrubs have been there, so that might not be the best solution. RCW 7.28.083. They can do this even though they weren't the ones who owned the property at the time. And if they sue you for that, you'd have to pay their legal fees, too, under the same statute.
It's important that you are 100% satisfied with my courtesy and professionalism. Thank you.