GLYPHOSATE! (SP?)
Commonly sold under the trade name "Roundup".
This is applied to the green growth and makes its way down to the roots where it stops them producing food or the plant. If it comes in contact with the soil it is neutralised, so it doesn't poison anything but the plant it's applied to.

It's more commonly used to control things like nettles, but I have used it to kill off unwanted bamboo plants (I'll have the Bamboo Forum flaming me now!) by painting the leaves a few times with the stuff.

If you paint the green parts of the offending tree several times,this should do the trick for you. (I'd recommend doing this at night, otherwise standing at the top of a ladder painting the leaves might JUST arouse some suspicion!)

Good Luck!
(and don't tell them I told you!!)