The name for the ‘vegetal’ parts of Islamic designs is variosly called ‘arabesque’ or ‘Islimi’ - the latter is used in the book ‘Arts and Crafts of the Islamic Lands, which I highly recommend. In any case, they are an important part of these patterns, that I have neglected until now …
This is the basic geometry (incidentally, it is the pattern used on the cover of the Broug book).
The final design was just to fill in parts of the pattern with shapes that are traditionally used (or my attempt at them). I drew them on the original graph paper and then traced them to get an even shape:
The colours are a little wacky, but the overall look is good, I think. Perhaps still a little simple, but a good first try.
The completed item is here