By nature, some trees reproduce themselves by sending up new growth from underground. Aspens (Populus tremuloides) are one example of a desirable native suckering tree species. Tree-of-heaven (Ailanthus altissima) is an example of an undesirable exotic suckering species. Neither is a particularly long-lived tree, but both grow quickly and can create havoc by sending up new shoots. Other trees that commonly produce suckers are fruit trees, such as apples, American beech (Fagus grandifolia) and sweetgum (Liquidambar styraciflua). Many trees that don't typically sucker will send up suckers if they are cut down or severely pruned.
Many invasive trees produce seeds carried by wind and water, or dispersed by birds that feed on their fruits. Two notorious exotic invasive trees are the Melaleuca (Melaleuca quinquenervia) and the princess tree (Paulownia tomentosa), which foresters estimate produce a million or 2 million seeds per tree, respectively.
Sometimes trees invade by growing so rapidly that they not only outgrow their welcome, but produce only weak structures that are very vulnerable to damage. Bradford or Callery pears (Pyrus calleryana) and silver maples (Acer saccharinum) are examples of fast-growing trees that tend to have short lives and go out with spectacular splitting of trunks or breaking of branches during ice or windstorms. Invasive tree roots are another form of greedy growth that creates competition with other plants and may cause problems with sewer lines, septic fields or sidewalks. Willows (Salix spp.) are water-loving trees that can invade septic fields or sewer pipes. Norway maples' (Acer platanoides) dense, shallow root system out-competes most anything you try to grow under them and can raise driveways or sidewalks that get in the way.
Some tree species may invade your landscape in multiple ways. For example, tree-of-heaven can reseed as well as sucker, and the princess tree can sucker as well as seed aggressively. Silver maples and willows can damage pavement, as well as wreak havoc above ground. Regardless of how they operate, invasive exotic trees have no place in the landscape; native invasive trees need careful placement. You'll save yourself hours and dollars by learning more about a tree before you include it in your garden.