You may have some of the tree roots in that bed, depending on how close the trees are, but plants growing in soil put out lots of roots that search for moisture and nutrients the soil has. The only way to know for sure is to take a spade and cut all around the edges of the planting bed to se just where those roots came from.
Many years ago I thought that pulling up the plants that once grew in a garden bed, hauling that to the compost pile, composting that material, and then hauling the finished compost back to the garden plot was too much work, so today I leave the plant material that grew in that planting bed there and cover it with finished compost and the soil bacteria digest that material right there putting the nutrients the plants took from that soil right back into that soil along with what is in the compost.
Some people, that still hang onto the clean garden bed concept, cut off the top growth because someone told them they had to remove all the plant material or they would get plant diseases but they leave the plants roots in those beds (mostly because they cannot see the roots) so if there will be any plant disease the roots will have them. I have found over the years that leaving all the plant growth in a planting bed, and getting the soil of that planting bed into a good, healthy condition will do more to prevent any plant disease then anything else you can do.

Thanks kimmsr. I'll try cutting around the box to see where the roots originate. That seems like a good idea.

You also make alot of sense about leaving the planting material right where it is. It's too late this year, but I'll give it a try next year.

Kimmsr, I'd like to try the leave the dead plants in the bed and add compost method. I am wondering, can you do that with asparagus, too? I've been told that it's best to burn the plant in the bed after it turns brown.

Burning an asparagus bed is really old school thought and has not been suggested for many years because the burning removes more valuable nutrients than it provides. After I stop harvesting the Asparagus from my bed I will put some compost on that bed and plant a summer cover crop (or 3) which is knocked down to plant a winter cover crop, and the Asparagus ferns are knocked down once they turn (which they did this week) In the spring when we start to harvest the Asparagus I will stomp on the winter cover crop (Winter or Cereal Rye) to stop the growth and allow the stalks to lay on the soil as a mulch.