It depends on the lease. Generally, the tenant is responsible for maintaining the house (and garden) . . . this is no different than the tenant washing the floors and taking out the trash. Otherwise, one should be living in a high-rise that has maid services.
However, if there is a prized garden, the landlord may want to write into the lease that a gardner will do A, B, C . . . once a week/monthly . . .
Now, if you find a larger expense -- i.e., a large tree that needs to be pruned, this would be like if the house needed to be re-roofed -- the landlord should be responsible for pruning and taking the limbs away.
Check out your local rent board/authority.