Did you build the raised beds on top of the root area for the trees? If so,that's part of your problem. The tree was suffocating and needed to establish new roots quickly. In this case, you'll have to allow the tree to establish at least a few roots in your bed if you want the tree to survive.

If you didn't build on top of the existing soil, the tree roots will still grow in to this nice loose, breathable soil. It's what roots do :-) In this case you can just cut or remove the roots as you find them. Severing them underground with a shovel or spade works too. Completely preventing the roots from growing into your raised bed will require the installation of a root barrier (a strip of plastic placed under the soil).

Some trees are allelopathic - they secrete a substance that inhibits other plant growth. Black walnut is one such tree (secretes juglone). If this is your situation, you'll have to plant juglone resistant plants in your beds, or you'll have to remove the tree.

BP

forget raised beds. Instead create planters elevated 1 foot over the ground. This way you can have your garden and your husband can continue to enjoy his trees.

The planters might be a solution. They would have to be off the ground. Believe it or not, my friend tried planters on the ground and the tree roots went right up into her planters! We have cedar and pine trees around here trying to survive in an area with lots of rock and little top soil so I guess they seek out any source of soil. I may look for the root barrier as well. It will mean taking the soil out of the raised bed but worth a try. I have a very limited sunny area in which to plant so need every inch of space.

Hello Maggie. We have built extensive gardens here under the shelter of a hardwood tree forest - maple, beech, black cherry, etc. I just leave the roots where they are. If you look at the way plants grow in a forest, they grow in the most unlikely places, including on top of tree roots, and especially in the crooks of mature trees. I add lots and lots of compost and give hostas a handful of bone meal in spring. Our shade gardens are never watered, except when planted, and, if I do say so myself, are phenomenal.
We have also planted this way successfully under mature spruce trees.

Trees will take all the nutrients out of the soil and your flowerbed won't do well under them, unless you feed and water it regularly.

Again, I do not feed or water regularly. The gardens have been in place for 10 years. I have never had a problem with nutrients.

There are plants that can grow well beneath trees but you'd have to be more observant as to how to pair these up. For example, it's difficult to grow any large deep rooted flowers where it's already crowded with roots. So the trick is to find plants that can insert itself in pockets of area and so choose small young plants so that these can gradually insert itself. These must be plants that can in the long term fend for theirselves. As ninamarie mentions, the addition of compost does help such plants. I can tell you though that you can forget growing anything beneath pines or firs, or spruce as these do take up all the water and nutrients and not to mention, the shade they provide is a deep dark shade - too deep for most plants.

Plants that can grow beneath trees include ferns, hostas, astilbe's, geraniums, columbines, etc.. As a general guide, look for plants under the description of woodland plants or shade plants.

In BC for example, one sees a lot of moss, ferns and rhododendrons, (another plant comes to mind but the name evades me) growing beneath the cedar canopies. These are particularly suited to acidic conditions.

Now for many of our favorite sun loving, fertilizer loving blooms, it just may be difficult to grow them beneath trees, as runningtail describe, there will be competition for nutrients. - in such cases, a planter may be suitable.