Hello, welcome to the website. My name is Tony, and I should be able to assist you with this.
Can you clarify for me please whether the landlord is refusing to allow you to do anything with the trees concerned?
You could argue that "keeping in good condition and weeded" does not extend to cutting back large trees, and I doubt it was intended to mean that anyway. This type of clause is usually about keeping things tidy, and in good order, and removing trees or seriously curtailing them is, I expect, beyond the scope of that type of clause.