I believe the best herbicide for many situations may very well be patience and elbow grease.
Disclaimer: While I am an avid gardener and I enjoy learning about the agricultural sciences, I am a computerologist, not a botanist, biologist, or chemist.
I don't know all there is to know about herbicides. I know there are broad-spectrum herbicides and selective herbicides, but I am only aware of two of them by name.
There's the well-known broad-spectrum herbicide glyphosate, of course, but it is absorbed by foilage and has low persistence in soil. This is why the application instructions typically say to apply in sunny weather when the winds are light and rain is not expected for several days.
Then there's tebuthiuron, another broad-spectrum herbicide. It's been almost fifteen years, now, but there was a famous case of landmark oak trees in Auburn, Alabama, being poisoned with Spike 80DF -- a specific formulation of tebuthiuron made by Dow -- by a rabid sports fan upset by the recent success of a rival. Because tebuthiuron is absorbed through the roots, has high persistence in soil, and inhibits photosynthesis, the trees were assured to die. My understanding is that this is an industrial herbicide, though, and may be difficult for consumers to obtain. It's banned in the EU.
Don't do what I did, which is to throw caution to the wind in frustration.
I was successful in killing a tree under my roofline. It was planted less than a foot away from the foundation and was difficult to prune due to its location amidst yews and aborvitaes. The trunk grew to about 3" in diameter before I decided to cut it down. I cut it down to about an inch or two above the ground. It was nothing if not insistent, though. It continued to put out new shoots and I would cull them and/or spray it with glyphosate when I noticed it. After a couple of years of this dance, I escalated things. I cut the trunk down beneath the surface so that no part of it was above the ground. I doused it with a number of chemicals based on advice I received. Eventually, it stopped putting out new shoots. Over the next few years, though, the pair of healthy, well-established arborvitaes about ten feet away on either side began to brown and die. I think it's quite likely that whatever I did to rid myself of the tree also contributed to the demise of my arborvitaes.
If I had to do it over again -- having the benefit of hindsight as well as the patience that comes with age -- I would have tried to suffocate and starve it instead of saturating the soil with something that may hang around to impose unintended consequences. Perhaps I would have had success cutting it beneath the ground and capping it with a sturdy container impermeable to light, water, and air. I would have periodically removed the cap long enough to eliminate any new branches or shoots that should threaten to breach the barrier. I would have done this until it exhausted all of its energy reserves trying to reach the light.
Cellulose is constructed from chains of glucose -- probably the most common carbohydrate produced via photosynthesis. Plants synthesize carbohydrates, store them in tissue, and later metabolize them via respiration. If one is both diligent and patient in limiting the plant's capacity to photosynthesize and respirate, I am sure it will eventually die.
Glucose photosynthesis: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂
Aerobic respiration: C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + ATP
Good luck with your tree!