depends on what kind of trees, how big they are, how deep/tall the bed is, what you want to keep from the existing bed, and how deep you want the border to be.

some trees deal well with root pruning. some don't. some put out 'sucker' roots into mulch that can be trimmed away. more get crabby when you do that.

I don't think any bed should be 8 feet deep - but I like being able to reach into the center of the bed without crawling into it, and my beds are only 3' deep, with one bumpout to 5' (I have a small yard, so they're proportional)

posting a layout would help- makeing up a layout might be all you needed to answer your question for yourself- things aren't so complicated when you're looking at it symbolocally, and a 3" circle is easier to contemplate, and move, than a 3' plant.

The tree roots will not be happy if mowed.

Gentle, swooping curves give visual pleasure (unless you prefer very formal 'scaping, in which case square edges might be more comfortable). I prefer encircling the trees with a light mulch [of 2" chips over 1/2" newspaper] out to and a bit beyond the dripline. You might be able to enclose two or three trees, and I've seen this very attractively done, with a stepping stone path laid on top of the mulch to a couple chairs or a bench, with shade-loving plants in colored pots to enhance the setting. Yes, if you can, swoop the tree circles' edging over to the fence. Try laying a hose or using some flour (or lime) to figure out what curves will look best. Flour-circle the tree driplines first, and then draw more curves to join with the fence bed, which itself will look better with a curve or two. Move the old edging if you like it, or find an edging you do like -- I'm not fond of plastic but it's much better than nothing, especially when I've blown my budget on plants, lol. BTW, if you can afford it or are handy with masonry, an edging that has a 3to5" mower edge (flat to the soil) is a wonderful timesaver. After marking the curves you like, place the edging along the drawn line, then finish by using several layers of newspaper (1/4" to 1/2" think when wet) to form a base, with a mulch of chips or shredded leaves on top of the paper. Be sure *all* paper is covered. Chips look very neat in a woodsy setting and will last a couple years before replenishing is needed; shredded leaves look nice but will gradually decompose and need to be refreshed once or twice a year. Most perennials are quite fond of a paper/shredded leaf mulch (hey, it's fresh compost right at their roots along with friendly worms!) and you will find weeding becomes an almost rare chore. The semi-exception is with plants that demand great drainage... give lavender, rosemary, dianthus, phlox, etc. a mulch of coarse sand from their stems to a couple inches beyond their dripline instead of the leaf/paper mulch. Many trees and shrubs like a pebble mulch from their trunk to 6" out, and then paper/chips to the dripline. Those pebbles also discourage mice from nibbling on the bark during the winter.

Mulch is the secret to having a large garden with little work :)

I know that I'm not answering the question that you asked, but: Have you considered putting in a tough, reasonably weed-suppressing groundcover in the forward part of the bed, instead of replacing it with grass? Then you wouldn't have to put anything around the trees, their roots wouldn't have to deal with a mower, and if you get more ambitious in your gardening later, you won't have to strip out grass.

I realize that this may still be too much work (the ground cover would still need some weeding) but I wanted to mention it as an option.

Burnet

Trees planted in parks have grass all around - and the mower comes by regularly. If you were to take the catcher off when you mow in this area then there would be a frequent top-up of mulch for the trees, in addition to any feeding you might provide.

Some trees sucker badly (elm, for example), or have knees sticking out of the ground (swamp cypress). For those, a surface of mulch saves on lawn mower blade wear and tear.

Unless your garden design needs to have a defined border for the trees you may not need one. Just let the grass taper away in the shade zone near the trunks.

Obviously I don't know what sort of fence you have but, instead of pushing the gardening area back, you could create a path along the fence, maybe even with a pergola over it. That way you can access the fence for maintenance and painting (if needed) and you can access both sides of the bed without stepping on it. Much easier for feeding, trimming and all those close-up chores.