What is an Invasive Exotic?
There are hundreds of species of flora and fauna known to be harmful when introduced beyond their natural ranges. Seven trees in the United States are considered invasive and listed as an alien exotic pest.
According to Invasive.org , "any species, including its seeds, eggs, spores, or other biological material capable of propagating that species, that is not native to that ecosystem; and whose introduction does or is likely to cause economic or environmental harm or harm to human health" should be considered an invasive.
Not every tree you plant is desirable and can actually be harmful to a particular location.

The Unwanted Tree List
You need to be extremely careful when planting these species outside their natural habitats:

Black Locust ( Robinia pseudoacacia L.) yellow locust, locust
Black locust has been planted extensively for its nitrogen fixing abilities, as a source of nectar for honeybees, and for fenceposts and hardwood lumber. Its commercial value and soil building properties encourages further transportation outside it's natural range.
Black locust is native to the Southern Appalachians and the Southeastern United States. The tree has been planted in many temperate climates and is naturalized throughout the United States, within and outside of its historical range, and in some parts of Europe.
Once introduced to an area, black locust expands readily into areas where their shade reduces competition from other sun-loving plants. The tree poses a serious threat to native vegetation in dry and sand prairies, oak savannas and upland forest edges, outside of its historic North American range.

Chinaberrytree ( Melia azedarach L.) China Tree, Pride of India, Umbrella Tree
The Asian Chinaberry is a small tree, 20 to 40 feet tall with a spreading crown. The tree has become naturalized in the southeastern United States where it was extensively used as an ornamental around old southern homes. It has managed to spread by root sprouts and an abundant seed crop. It is a close relative of the neem tree and in the mahogany family.
Chinaberry's fast-growth and rapidly spreading thickets make it a significant pest plant in the U.S. Even so, it continues to be sold at some nurseries. Chinaberry outgrows, shades-out and displaces native vegetation; its bark and leaves and seeds are poisonous to farm and domestic animals.

Mimosa ( Albizia julibrissin Durazz. L.) silk tree
Mimosa was introduced into the United States as an ornamental from Asia and Africa. It has escaped into fields and waste areas and its distribution in the United States is from the mid-Atlantic states south and as far west as Indiana. It is a flat-topped, thornless, deciduous tree which reaches 35' in height. It can sometimes be confused with honey locust because of the bi-pinnate leaves of both.

It does not establish in forests, but commonly occurs on forest borders. It can also invade riparian areas and spread downstream. It is often injured by severe winters. According to the U.S. National Park Service, "its major negative impact is its improper occurrence in historically accurate landscapes."

Royal Paulownia ( Paulownia tomentosa ) Princess Tree
Princess tree was introduced into the U.S. as an ornamental and landscape tree around 1840. Paulownia has a rounded crown, heavy, clumsy branches, reaches 50 feet tall, and the trunk can be 2 feet in diameter. The tree is now found in 25 states in the eastern U.S., from Maine to Texas.
Princess tree is an aggressive ornamental tree that grows rapidly in disturbed natural areas, including forests, streambanks, and steep rocky slopes. It easily adapts to disturbed habitats, including previously burned areas, forests defoliated by pests (like gypsy moth) and landslides and can colonize rocky cliffs and scoured riparian zones where it may compete with rare plants in these marginal habitats.

 
Tallow Tree ( Triadica sebifera (L.) Small.) Chinese Tallow Tree, Popcorn-tree
The Chinese tallow tree was purposely introduced into the southeastern US as early as the 1700s.
It comes from China where it has been cultivated for about 1,500 years as a seed-oil crop. In the US, it is associated with ornamental landscapes as it makes a small tree very quickly
The tree is medium sized tree growing to a height of 50 feet, with a broad pyramidal, open crown. Most of the plant is poisonous, but not to touch. The leaves somewhat resemble a "leg of mutton" in shape and turn red in autumn.
The tree is a fast grower with insect inhibiting properties. It takes advantage of both of these properties to colonize grasslands and prairies to the detriment of native botanicals. They rapidly turn these open areas into single species forests.

Tree-of-Heaven ( Ailanthus altissima (P. Mill.) Swingle) ailanthus, Chinese sumac, stinking shumac
The TOH was introduced into the United States by a gardener in Philadelphia, PA, in 1784. The Asian tree was initially promoted as a host tree for moth silk production. It rapidly spread because of it's ability to grow quickly under adverse conditions. It also produces poisonous "ailanthene" in bark and leaves that helps limit competition
It now has wide distribution in the United States, occurring in forty-two states, from Maine to Florida and west to California. It grows stout and tall to about 100 feet with a "fern-like" compound leaf that may be 2 to 4 feet long.
Tree-of-Heaven can't handle deep shade and is most commonly found along fence rows, roadsides, and waste areas. It can grow in nearly any environment that is relatively sunny. It can pose a serious threat to natural areas recently opened to sunlight. It has been found growing up to two air miles from the nearest seed source.

White Poplar (Populus alba L.)
White poplar was first introduced to North America in 1748 and has a long history of cultivation. It is chiefly planted as an ornamental for its attractive leaves. It has escaped and spread widely from many original planting sites.
White poplar is found in forty-three states throughout the contiguous U.S. Click here to see another distribution map.
White poplar outcompetes many native tree and shrub species in mostly sunny areas such as forest edges and fields, and interferes with the normal progress of natural community succession. It is an especially strong competitor because it can grow in a variety of soils, produce large seed crops, and resprouts easily in response to damage. Dense stands of white poplar prevent other plants from coexisting by reducing the amount of sunlight, nutrients, water and space available.