Under most circumstances, if your neighbor destroys a tree located on a common boundary, they will be responsible for any damages you suffer.
If your neighbor cuts or removes branches or roots of a tree bordering both properties, your neighbor will not be responsible for any damage that may result to the tree.
Generally, if your neighbor is found to have cut down or destroyed a tree you shared, your damages are measured by subtracting the market value of your land after the tree was destroyed from the market value of the land immediately before the tree was destroyed.
If one of your neighbors has informed you they will be destroying trees that you share against your objections, you may go to court and file an injunction. Although these injunctions can be hard to win, courts have prevented neighbors from cutting, destroying or injuring a row of shared trees that provided the other neighbor shelter from outside elements such as wind.
An attorney experienced real estate lawyer will be able to advise you of the rights and remedies available to you if your neighbor has damaged a shared tree. If you have been accused of damaging a tree, a lawyer can ensure your interests are represented.