Paulownia tomentosa, invasive non-native tree from Asia.
Those "nut like" fruits are actually seed capsules that contain hundreds of tiny seeds. When the nut cracks open, they are released and carried by wind and water to new places.

A single tree is capable of producing an estimated twenty million seeds

Here is a link that might be useful: Link to info

Could be the wisteria tree? Wisteria is in bloom now. Do you have a picture?

There is no such thing as a "wisteria tree". Wisteria is a vine that grows up into trees or sometimes is trained on a standard to resemble a small tree.

Large bell like blooms, and has some sort of nut like fruit.

Definitely not wisteria which has pod-like fruits because it is a legume.

Yes, it's the Paulownia, but I differ in that at least here, it's not the wildly invasive tree everyone states. I very occasionally see one and you might think with so many seedlings, they would be everywhere, but no. This is the rural countryside by the way.

I think they are indeed lovely but the very size of the leaves- huge- and the frequent dieback of large trees, would discourage me from ever planting one. And, of course, horticulturally I would feel responsible....

Well: I have one growing at the edge of a woods. It is clearly visable from our front door and it is stunning. I have been here eight years. Several years ago, I was looking at the detail of the fallen flowers, and spotted a two inch tall seedling seemingly growing right out of a flower. I left it alone and watched it grow until winter when it had reached approx. 12 feet! Very impressive! Last year I also let another seedling grow that I thought was a super large variety of okra because it was growing where I had left an extra 4 pak of okra. There are also several growing within a foot of the main tree. I suspect that they are growing from the roots, but they could be seedlings. They will be cut down the next time I have a saw in hand.

I have no idea how many seeds are actually expelled, but considering I have only seen two confirmed seedlings in eight years, they must be the most ineffective seeds ever designed.

Invasive progress sometimes takes a more exponential effect. It has been stated that Japanese honeysuckle was sold in garden centers for 80 years before it "started" to become invasive. Now, in the southeast, Japanese honeysuckle is present in more acreage than any other non-native invasive plant.

I think what happens is that a low number of individuals (relative compared to the number of seeds distributed as chas045 says) gets out there until one day you have just enough to flood the area with billions of seeds. Then the explosion occurs and you have your invasion.

Then people say - "if only we knew it would do that we could have cut the early ones down and stopped it." As the climate continues to warm, these seeds will find more and more acceptable climate for germination. In my area, the number of mature blooming paulownia increases every year on the roadside. More blooms = more seeds. Soon the invasion will be in place. And then it will just like Chinese wisteria or Chinese privet or Japanese honeysuckle - too big a task to tackle.

Western NC has a serious invasion problem with this tree, even up into the mountains. Drive through there this time of year and you'll see.

"Now, in the southeast, Japanese honeysuckle is present in more acreage than any other non-native invasive plant."

I am wondering about the term invasive and non native as well. Isn't the real problem only for those plants that don't play nice with others around them? From what I see, that would apply to kudzu and perhaps wisteria. I hear that wisteria kills trees, but I haven't noticed that all those trees near Chapel Hill beautifully covered with it are dead. I am also curious about honeysuckle. I do see distorted little trees that are twisted by the vines. However I see that sweet gums and winged elms that I suppose are native must have evolved over mellinnia, long before Japanese hunneysuckle showed up, to defend against the native variety.

The Chinese Empress Tree doesn't seem to be offensive. Yes, it is becoming established, just like all those pesky pines, big old oaks, elms and whatever. How about all those nasty little Chinese gardenias and azaleas attacking everyone's front yard?

Two things to consider, as you said:

Non-native: not all non-native plants are invasive. Many happily sit in gardens (like gardenias and asian azaleas and camellias). Why not have them, you say? Probably the best reason is that they offer no ecosystem services to the local environment. Insects and the critters that rely on insects (notably birds which feed insects exclusively to their babies) evolved with local native plants. Monarch butterflies are probably the most notable examples of insects that have a special relationship with a certain genus of plants (milkweed). But most other insects have a similar exclusive arrangement with another native plant. Reduce the native plant population and you reduce the native insect population and then that in turn reduces the bird population.

Now that we have destroyed most of the wild land (homes, roads, businesses have been developed), our yards and our roadsides are some of the few places left for native insects to find their host plants.

Invasive plants - only a small group of non-native plants become invasive, but they can do a lot of damage. When japanese honeysuckle comes in, it doesn't just distort little trees. It's vegetation out-competes native plants (especially young and small plants) for natural resources like water and light. Yes, wisteria probably only kills a few trees, or it certainly takes a long time for it to weaken them. But again, what is happening is that once it leafs out in the spring, its dense foliage prevents light from reaching young and small plants, so they wither and die.

So those pesky pines, big old oaks and elms are performing valuable services for the local environment by sustaining local insects and every creature that marches down the food chain from insects. The Paulownia is not offering any services - the insects that use it for a host plant got left behind in Asia. And everywhere there is a paulownia in the wild landscape, there is one fewer (or more) native plant for the ecosystem.

Hope that helps.

Here is a link that might be useful: More info here

Ditto esh_ga.

Another non-native invasive around here, in the Piedmont of SC, Is those darn Bradford pear trees.

I have them all over my pasture. I cut them down and they come right back. Time for some roundup!

Here is a picture of the tree
Sorry such poor quality...never take pics while driving!

The worst on my property is honeysuckle, privet, and multiflora roses. I have been fighting a battle for over 30 years.

I cut mine back after the flowers fade. It keeps them from dropping seeds. They will then shoot back up from the ground with large leaves and almost 10 feet tall!

Here is a link that might be useful: Arthur in the Garden!