Personal property is included as part of the coverage in a standard homeowner's policy, and foliage could be considered as part of your personal property. Make certain to itemize the trees and shrubs that should be covered in your home inventory, providing pictures or video to specifically identify each. But since you mentioned that they are expensive trees and shrubs, it might be best to give you a little more information regarding personal property.
By default, a standard homeowner's policy will only cover personal property up to a percentage of the value of the policy. In many cases, the family's property will already have a higher value than the covered amount, even before you calculate expensive foliage into the home inventory. You can increase the amount of personal property protection or take out a rider policy that increases the coverage. Regardless of the options you use to insure the trees and shrubs, save your receipts and provide documented evidence of the size, type and condition of each item.
If one or more of your trees dies, the value of the tree may be covered under your personal property portion of the policy, but removal of the dead foliage may not. For example if the tree is valued at $1000, and is later hit by lightning and killed, replacing the tree will be at least partially covered under a standard policy without riders, but the policy may not pay for more than a few hundred dollars of the removal cost.
Your best option is to speak with your insurance company or a dedicated representative about your concerns. You may be advised to take out a personal property rider along with increasing the coverage for tree removal, or you may be offered a rider specifically written for your situation which includes both the loss of the foliage and its possible removal. A rider of this sort has the advantage of reserving the personal property coverage portion of your policy for things such as you furniture, appliances, and other property.