If you want to remove the tree at your own expense, you can propose that to the landlord. You can also offer to get estimates for tree removal for your landlord to review and potentially approve.

The reasonable amount of time, alas, is exactly that -- an amount of time that is reasonable under the circumstances. Factors such as the risk to the structure from the tree, the level of inconvenience to you, the amount of time it takes to schedule the contractor to come out and remove the tree, and the like, all factor into what is reasonable.