Answer: In spite of the childhood song, mulberries are trees, not bushes, and can easily gain heights of 40-70 feet. From your descriptions of the trunk that requires a chain saw, I am guessing that the previous owners also tried unsuccessfully to get rid of the trees by repeated chopping.
There are two main types: red and white, and although the red is native and the preferred of the mulberries, either may or may not be right for your yard. Because of the heavy fruit set, mulberries are prolific seeders, sprouting new trees wherever they can. In the right place they can be a nice addition to a backyard since they are a favorite of birds. In fact, in the spring when the fruit is ripe, the trees will chatter with the talk of happy birds! Personally I love the mulberries that have been wild-planted by the birds on my land. That said, and since you are considering a border along the fence line, you may want to keep a few to grow into food-trees for the birds. A mixed border of shrubs and trees, both deciduous and evergreen, is a very inviting place to all sorts of birds!
If that's just not in your plans, then possibly the only way to tackle the large trunks is with brush killer which is available at hardware stores or nurseries. Cut back as much of the tree as you can, and carefully apply the poison to the tree with a brush according to the directions. It is very toxic, so be sure keep it away from plants that you don't want damaged! Read the label carefully.
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