I live in the UK and we have the full gamut of seasons. From freezing January to sweltering August and as my flat faces South, plenty of sun from May to September.
It must be something she can climb.
Do some shopping around! I'm not sure what your budget is or how much you're willing to spend, but real indoor cat trees -- quality ones -- tend to run upwards of $100.
While you probably won't be able to grow a tree inside that's big enough for your cat, you could go for a substitute instead. I had a friend when I was younger who had an enormous branch in her house for her cat to climb around on. It was set at an angle, so it wasn't straight up and down, but leaned against the wall and gave the cat fun places to perch. She even built a few platforms into it where the cat could lounge.
Any large hardwood branch should work for this, just make sure it's not weak in any places before you bring it in.
Easier still would be a store-bought carpeted cat tree.
It sounds like you're asking about a live plant that your cat can climb into. I don't think you're going to find anything. The reason being, that no matter how bright the inside of your house is, you most likely won't have enough light or space for a tree to grow to a thickness that can support the weight of your cat climbing through it. Outside of a mall, I doubt you've ever see shrubbery that big inside of a residence. I know I haven't.
However, if you simply want your cat to have access to living plants, you could get a regular cat tree and simply place plants at the base and even in the tree. They also have cat trees that simulate real trees. I think that will be the best option to achieve what you want.
I simply don't think an indoor tree is going to get much more than broom handle thick. Which won't support the weight of a cat.