[Qgis-user] How to create polygons from areas of overlap?

[Martino]

I have a vector layer containing polygons that represent vegetation
patches. There are 15 vegetation types (and most have more than one
polygon; i.e. the polygons are multipart). Some of the polygons overlap
as the areas have more than one classification (e.g. a polygon
representing one plant species and a polygon representing another plant
species). 

I would like to produce a new layer in which all of the overlapping
areas become new polygons. For example, an area of overlap of a patch
of Species A and a patch of Species B would become a new polygon
representing areas containing both species (and the non-overlapping
areas would become separate polygons).

So I start with 

Polygon with Species A
Polygon with Species B

and the result would be

Polygon with only Species A
Polygon with only Species B
Polygon with Species A & B

I am unable to figure out how to do this!

If anybody has any suggestions, I would welcome them.

With thanks.