Whatever you do, don't use a tire. Tires, being rubber...can transfer much heat to the wood of the tree.
I can see where your choice might be harder to come by....but an old tire should not---in my opinion, be one of them.

I know people have used them....generally, they are seen at an angle, used as a container with plants in them usually out on a lawn or at edges of gardens.
Do keep in mind good drainage.

I think she was kidding. I sensed irony in her plea. ;-)

I seem to remember seeing terracotta ring segments similar to what you are wanting, but don't remember where. It seems to me, one of the reasons they are not more readily available is it would be difficult to produce the variety of sizes required to serve the market. I have found a few other solutions.

http://www.eco-gardening.com/612/copper_tree_ring_planter.php

Three of these could be arranged around your tree so that the trunk comes through the center:
http://www.markstaar.com/product/129/789.html

http://www.markstaar.com/product/129/790.html

Also, if you find an air conditioning duct maker, some other metal shop, fencing company, etc., you might be able to have something custom made to your liking.

Good Luck.

Alfresco Home Cobblestone sells them. However I find that buying round or oval galvanized tubs (3 if a large tree) and cutting them with metal snips to the size you want and then clear caulking them together when you have what you want - Bueno! but raise it with decorative feet alittle when done so not blocking what the tree needs. INSIDEoUT designs Savannah GA.

I have a table built around a tree that goes through my deck and I wanted flowers around the tree. I found plastic faux pavers at Walmart. Turned them upside down and planted away. Each end is round to fit the next piece together. It took seven to complete the circle large enough to go around the trunk.

Prescott AZ