Excerpted from Purdue University - Ask Extension :

Whether eliminating black locust competing with oaks in a tree planting, or controlling invasive woody shrubs like autumn-olive or Asian bush honeysuckles, cutting the stems near ground-line and treating the stumps with an appropriate herbicide provides a one-two punch to the target plant. Severing the stems from the roots cuts off the flow of photosynthesis-produced food to the roots, and may prevent production and ripening of seed, if done early enough in the growing season. The stump left behind provides a small target area for herbicide application, limiting the damage to nearby native plants that can result from foliar herbicide overspray and drift. The herbicide application should kill the roots and limit re-sprouting. While it requires two sequential operations, cutting and spraying, it may be one of the most efficient methods to control many smaller-diameter woody plants due to the high rate of control achieved when done correctly and limited damage to surrounding desirable vegetation.

So how do you do the “cut-stump” treatment correctly? Several factors need to be considered, but some common elements to effective and efficient treatments include a cutting tool you can operate safely, effective herbicide type and application, and correct timing for the site and situation... [know how to operate power tools] safely and wear appropriate safety gear. I prefer brush saws for near-ground cutting as it limits bending over and is safer to operate than a chainsaw, but they will struggle with some larger diameter or denser stems a chainsaw could cut easily. Handsaws, loppers, pruners, or blade-tools like machetes and brush-hooks are hand-tool options, but larger stems will require more time and effort to cut or will be too large for some hand-tools to handle. 

Herbicides should be applied immediately after you have cut off the stems. A good choice for many applications is herbicide containing the active ingredient Glyphosate. Glyphosate concentrate herbicides containing 42% or more active ingredient are widely available at many garden and farm stores. Read the label before using the herbicide to confirm the active ingredients, concentration, and requirements for safe and effective applications, including the appropriate safety gear during dispensing and application... For stems over 3 to 4 inches in diameter, you can apply a band of spray around the perimeter of the stump so the bark, cambium and some sapwood is treated. If you wait to apply the spray to cut surfaces a dozen minutes to hours after the cut is made, two things will happen that can impact the effectiveness of the spray. The surface of the stump will seal as it dries, preventing the entry of the water/herbicide mixture into the plant. You may also have difficulty tracking down the cut stumps if the area is weedy and brushy.

Some cautions about timing... If the stem is frozen, usually 25 degrees F or colder, herbicide may not penetrate and evaporate off the frozen stump. Also avoid applications when rain could wash off the herbicide within two or three hours of application.

Several other herbicides are available for this treatment including Triclopyr and 2-4-D/Picloram herbicides. Be sure to read the labels and understand the safety requirements and application instructions before using.

As soon as the tree is cut down, regardless of the season, it’s important to act immediately. Simply paint a brush-killer type
herbicide on the newly cut stump so a fair amount is on the ring of cambium all around the outside just under the bark. If the tree has been cut down for any length of time, you should re-cut the stump all the way into live wood before applying the herbicide.
The herbicide is absorbed and transferred to the roots where it should cause death. This may need to be done several times before the plant dies, but you should eventually get it. This procedure can be done any time of the year.

Suckers that keep coming up from the root system of a cut-down tree can also be treated with a brush killer. If the suckers come up in a lawn area, you’ll have to spray them with an herbicide that will be absorbed by them but will not harm the grass. Sprouts coming up in an area where you don’t need to be concerned with
good plants can be treated with a non-selective “brush killer” herbicide which kills or damages all treated plants. This herbicide application to suckers is probably best done when they are in full leaf and actively growing.

Finally, if you don’t want to use chemicals you can
eventually kill the plant by continuing to cut the
sprouts and suckers off. If you keep ahead of it, it will
eventually exhaust its food reserves and die in much
the same way as with girdling. However, with this
method a very large tree may take over seven years to
kill. Being constant with removal of the sprouts is the
key.