What plant does vanilla grow on? ...

Part 2 of the rodent problem!?
It grows on a tropical vine.

I hold rather court patio garden...?
This seem to be the answer:

http://en.wikipedia.org/wiki/Vanilla_(or...
When do you pick tomatoes rotten the vine...
Answers:    It comes from a climbing orchid, a tropical species -- scientific nickname Vanilla planifolia, but generally specified as West Indian Vanilla.

The bit that gives the flavouring is certainly the seeds, though the pods they grow within are also very fragrant. The pods open green, but are dried for use as a flavouring and perfume, when they turn blackish-brown.

Gorgeous stuff! Used to buy it direct from the growers when I was travelling contained by Bali; miles better (and a fraction of the price) of pods available here in the UK.