No one answered my question. If anyone is interested, I made it 8 inches deep. Hope it's sufficient depth.

Ok you want an answer. You are going to kill your tree. Piling 8 inches of dirt against a tree and putting down cardboard will ensure the tree gets no moisture or nutrients. Besides a tree should never be lower in the soil than the flare--the widening on the bottom or rogue roots will girdle the trunk and kill it.

Perennials do just fine under a tree if given lots of moisture and care for the first couple of years until it's roots are established. I have a huge hosta, a bleeding heart luxuriant and creeping jenny moneywort under my 30 year old maple and they do just fine, but I gave them lots of care and attention for the first 2 years

The dirt is not against the tree. It's on the other side of the sidewalk. The tree is in the strip between the street and the sidewalk. The bed is at the edge of the lawn where it meets the sidewalk. The bed is probably six feet away from the tree, with a sidewalk intervening. It is still under the tree canopy, though. It's about 30 sq. ft. How deep should the dirt be for the flowers? Or do you avoid any sort of raised bed for this situation? I haven't planted anything yet.

In that case your bed should be just fine. A raised bed is just fine in fact good. It is far enough from the tree to cause no problems. I assume there is dirt under that raised bed so 8 inches of soil is enough. However cardboard is unnecessary in my opinion. Just plant your flowers and give them good moisture until they get established. Then you can treat them like any flower. If I can make a suggestion for planting. Try the Stella De Oro daylily It grows about 18 inches tall and has beautiful Yellow flowers with drooping foliage much like daffodils have. It will give a big flush of flowers first and if you dead head the flower stems when they die it will rebloom--just cut them off below the leaves. They are tough as nails.The bonus is that they are fast growing once established and you can divide them and make more plants

I forgot to add that an inch to 2 inch layer of mulch on top will go a long way to keeping your plants moist. Last year was a very dry year here and I didn't have to water my flowers very often because of the mulch. When you do water water deeply so the moisture gets down to the roots. The finger test is a good rule of thumb for watering. Move back the mulch and stick your finger in the soil. If it feels wet you don't need to water. It is also a good test to see how far down the water went when you do water

Thanks for your advice and suggestions. I had a pile of topsoil/compost (8 cubic yards, half topsoil, half compost) delivered that I needed for other beds and to fill up a pit where a maple in the back had been removed last fall (the stump was ground and I dug out the chips) so I needed to get the new dirt distributed where I was going to use it. For this bed by the sidewalk I dug down in the existing soil (put it on a tarp), put down the cardboard, then put the dirt on the tarp back, mixing in the new topsoil/compost (which is nice soft, dark dirt) so there is about 8 inches above the cardboard. The cardboard isn't going to last. I was just hoping it would keep out the tree roots long enough so the new plants can get established. In digging down I tried to avoid cutting roots. They are there and continue on out into the lawn. I cut an edge around the bed to keep out grass from the lawn. I plan on using mulch and will check out the day lily.