Tucson was faced with a freeze that lasted 3 days this year (2011) and it killed many tree's. In hindsight the worst effected was the Willow Acacia. Almost all died, I would estimate over 10,000 (our company has removed over 500 so far).
The biggest problem I have seen with the Dead Willow Acacia Tree's is them regenerating from the root. I have seen virtual jungles growing as a result of the tree's death. What do you do with that new growth?
Well first do not be fooled into thinking that it will ever be a tree again. Similar to untreated Sumac stumps,the Acacia will be a shrub. They are hard to kill, very aggressive root systems need drastic measures. A beginning treatment of copper sulfate is the place to start (this will not kill all, but should kill the stump). After that you will find that the small 3/4 inch surface roots regenerate and grow "weeds" all over your yard. This is hard to control, I recommend starting with a weed killer to see if you can get it under control. If that does not work a pick may be a good investment. the roots are often small and easy to remove, if you let the weeds grow 1 foot tall often when you pull them the root comes up too. If this does not work or you have an overly aggressive root that will not stop growing then it is time for drastic measures.
Drastic measures include using salt to change the PH Balance around the roots. Simply buy a big 40lb bag from your local hardware store and put about 1/4 bag in the needed area. Use some water to dissolve the salt into the soil. Wait 3 weeks and see how you did. A second treatment may be needed.
Warning... Excessive salt can kill surrounding plants and will ruin that area of soil for future planting. So use with care. Good Luck! and give us a call if you need any help with your tree removal.