How many trees are needed for forest industry in Cities: Skylines?

The trees themselves are irrelevant for your forestry industry. All that matters is the green color in the Resources map mode.

Forest industry in the purple rectangle would be a 50/50 mix of forestries and wood processing factories. They won't import (at least not much), but they will export processed wood products to your generic industry and to off-map.

Forest industry in the blue rectangle would be mostly wood processing, but there is a very light green color in the southeast and southwest, so you might see a few forestries there. But these would likely not cover the needs of the wood processors in the district, so they will likely import quite a lot of wood from off-map.

It's less about the number of trees and more about the density. Even 3 trees right next to each other is enough to start showing as a 'forest'. The more trees you have in close proximity, the stronger the resource is. If you plop some trees in the blank space between the trees at the top of your image, you should start seeing some forestry resources appear.

Edit: When you zone for forestry, they'll plant more trees in the area, so the number and size of the trees in the purple zone isn't really relevant since you've placed the industry there