This is a complex problem. The answer is: it all depends upon the type of utility and how it impacts your property.
1. Ask the utility for a copy of the construction plans. This will help you to determine how much of your property will be affected. How close will they come to the house?
2. Find your deed and determine whether your property is on metes and bounds (property defined by measurements from Range, Township and Section Lines, or if your property is on a platted lot in a recorded subdivision. With this information, you can go online to research the title and see if the utility actually has an existing easement across your property. If they do not have an existing easement, then they would have to buy one. You might request that they pay for the lost trees and fence as part of the compensation package.
3. What sort of utility lines will be installed? There are many different types of utility, including electric, gas, fiber optic cable, water line, sewer line, and even cable TV. The type of utility would determine how it will impact your property. Will there be multiple companies sharing this same utility easement?
4. If it is going to be an electric line, I suggest reading the Wikipedia entry on "overhead power line", so that you can anticipate what that might look like. If it is a water or sewer line, you might lose your trees, but it would not create an aboveground eyesore. You would probably not be able to plant trees that would grow roots into underground pipes.
5. If it is an underground utility, will it affect your septic tank? What about the alternate field? If it affects both, that might make your home uninhabitable, and perhaps the utility would have to buy you out.
As you can see, it is complicated, and very much depends upon the specific conditions on your property.