Strange Knoll Tulip Stool  

It is a Saarinen stool by Knoll except someone added the short back to possibly turn it into a vanity stool.  It is really all up to you to decide how you want to proceed with the re-upholster.  If the back is useful to you, keep it otherwise remove it if you do not want the back.  Keep the new fabric plain and simple.
see archival collection at knolltextiles.com
http://www.knoll.com/shop/by-brand/knolltextiles/upholstery/archival-upholstery
or similar good solid color plain weave upholstery by others or if for personal use, get creative with it using your own choice of pattern color and material.
Solid color vinyl or leather will work too.
Also for future reference, if you decide to keep the back, make sure it is noted that it is a later addition by the owner to avoid confusion.

Thank you for your help minimoma,
as the backrest is very comfortable, i will leave it as it is. As you said a plain and simple fabric fits best, especially on such a small furniture. The two websites are a great inspiration, i think i add some colour to my home and choose a brighter fabric, although a white fabric would transform this into a cool and sleek stool...hard decision^^

Hi Jay.
 
The back should be removed.  Mr. Saarinen would probably __________, and then roll over twice before having a cigarette.  
Sorry,  just my thought.
Aunt Mark

Hi Mark,
you are certainly Right 😀 It seems that i have no other choice than to go back to the original design as doing the reupholstery for this backrest version would be way to complicated for me.