Can you post a picture straight on of the whole plant. Doesn't look at all like a tree.

If this is being grown indoors, I will tell you right now that lavender makes a very poor houseplant. Indoor conditions are too dark. That's why no flowers.

It also requires a very fast draining and lean potting soil and only enough water to keep it hydrated.

Here's a picture of the whole tree. You can see all those brown leaves. The plant is currently growing in a pot on my balcony and I live in British Columbia. Will the tree survive outdoors during winter?

It should. Lavender is fully hardy in all but maybe the most interior portions of BC. But tree forms are somewhat less hardy than the more commonly grown shrubs so I might just bring it in overnight if you get any of those Fraser River Arctic blasts.