Are cheap trees a hardwood or softwood?

Price has nothing do do with it. Balsa is a hardwood - though it's very soft. Fir is softwood but is more expensive than pine which is more expensive than spruce (all softwoods). Two definitions for hardwood/softwood:

1) softwoods never lose their leaves/needles; hardwoods do when it gets cold.

2) softwood seeds fall to the ground; hardwood seeds are contained within something that drops from the tree (Apple, chestnut, acorn, etc.)