[recycled sixtie]

First off your landlord neighbour sounds very unreasonable. I would check to see if there are bylaws covering trees from neighbours encroaching your property. Here in Edmonton we have encroachment provisions in this regard. We have recently talked one neighbour into removing three large spruce trees because the roots came onto our property and it was getting hard to cut our grass because they were above ground. Also the roots were undermining our garage. At first he talked lawyer and then he must have consulted somebody and gave in and hired a tree cutter.

The other neighbour has a pear tree overhanging our property. The pears come down on our property and make a heck of mess. So in the past I have trimmed the branches that overhang our property. I am not sure if you could legally do that in your part of the world.

So my suggestions are:
Check LA City bylaws covering what you can and cannot do in regards to neighbours' trees.
Check tree encroachment procedure in your City if there is one.
Also if you sold your house now you have a legal problem with the new owners having an encroachment problem with the neighbours' tree so you want to get this problem solved now. I know you are not selling now but you want to get this problem solved anyway.
The last course of action is a lawyer as you want to get this situation resolved cheaply.
This landlord sounds like he does not want to spend the money. There must be an amicable arrangement to resolve this issue. Perhaps get the landlord's ph. number from the tenant? This situation is quite stressful for you but I feel confident that you will resolve it in an amicable manner!
Cheers, Guy