Girdling is a good way to kill the tree so it won't root sucker like a cottonwood, Russian olive etc. Takes a year to kill sometimes but is final. Can be done chemically it mechanically. Spraying the tree with the herbicide habitat will probably work too, double check the label. If you need it gone right now then chainsaw as low as you can go and paint the outside rings of the stump with concentrated round up. That part of the tree is the phloem and what carries all the nutrients, water etc throughout the plant, sort of like blood vessels in us. The inner structural part of the tree doesn't so applying chemical there is wasting it. Often pulling root suckering trees will break the root tips leaving them to reproduce. If they are a nuisance then I suggest taking them out when they are younger, than ones been around a while.
