During the ice age, the same plants grew as grow today. The glaciers stopped in Southern New York, the Northern border of Ohio and along the southern borders of the Great Lakes. South of there plants grew. Plants simply grew more to the south than they do today. The North Woods started at the edge of the Glaciers. Then came the firs and the maples. Most of South Florida would not have been tropical but subtropical. It would have had a few tropical islands where freezes would be rare. They would have reseeded South Florida with tropical plants at the end of the Ice Age. Cuba would have been Tropical. Birds from Cuba came to the Florida Keys after the end of the Ice Age.
So, as the ice melted, the plants of today moved north. The children of the ice age plants found the best places for them to grow and prosper. It was as if the plants moved into the area best suited for them.