Paulownia tomentosa

Paulownia tomentosa (Empress Tree) is a non-native invasive tree that grows rapidly.  The tree has HUGE leaves and can spread through either root sprouts or seed.  A root sprout can grow 15' in one season.  In the spring this tree will have pale violet fragrant flowers.  
 
This photo depicts the Paulownia tree and the very recognizable leaves and fruit.  In my experience, this tree can be easily controlled by using a battery operated drill and drilling a hole into the trunk of the tree.  The hole is then filled with a non-dilute solution of glyphosate like Round-Up.  The size and amount of holes that you drill into the tree depend on the size of the tree.  I typically will use a 1/2" drill bit, and then drill multiple holes depending on the size of the tree.  This slowly kills the tree and eliminates the root sprouts from forming.

 Northern Catalpa (Catalpa speciosa) looks similar to the Paulownia tree but is a native tree.  Both the bark and the leaf are very similar, the main differences are in the color of the flower--pale white as opposed to violet, the seed pods are long and bean-shaped and the lower leaves are not noticibly hairy like they are on the Paulownia. The Catalpa is pictured here with the white flowers. This tree has been known to grow rapidly and is fairly adaptable, but it is not invasive like the Paulownia.  Make sure you can correctly identify the tree you are attempting to remove to make sure that you are removing the correct tree.