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Personally I like catalpa trees if they are away from the house. I used to have a huge one in my back yard and people would come in the spring and ask if they could "harvest" the worms (caterpillars) that get on it in the spring, for fishing. They were never a nuisance (the caterpillars or the people - LOL), especially since the caterpillars only seemed to get on the catalpa trees. Also they aren't the type that make those big ugly nests either.
These trees do tend to be messy though. I call the leaves "paper bag leaves" because they are big and just plain brown. But I personally like the flowers in the spring, and I don't mind the "pea pods" either because they just aren't that difficult to pick up, and when they grow, like the flowers, they are interesting looking to me.
These trees remind me of when I was a kid because there were lots of them around my great grandmother's house. But when I had one, it never spread - I don't know why, maybe it was just because I kept the lawn mowed, I don't know.
If it came back VIGOROUSLY, with huge leaves after being cut back, it could also be a Royal Paulownia. They roughly resemble a Catalpa at first glance. They get fragrant, pale blue flowers when mature though, instead of the white flowers of the Catalpas...no cigar seed pods either. The wood of the Paulownia is sought by Japanese wood carvers, or used to be.
I like both of them. The RP is a fast growing tree.