[slow&rusty]

I do alot of this type of work for several companies and here are your only options from a legal \ common sense standpoint.

1) Get a copy of the legal survey for your MIL's property, it will have the Meets and Bounds Legal description and the exhibit showing the actual survey drawing of the property. This is typically kept in a safe place (under the bed or safety deposit box).

2) Hire a surveyor ($500ish) to read the survey from (1) and get him to stake, i.e mark the property lines or boundaries, he might get lucky and find property pins if he has an electronic pin finder. That way you can determine on who's property the fence is on.

3) If the fence is 50%/50% shared i.e on the property line between both houses (which I am sure is the case), approach the neighbor to determine how forthcoming they are about:
(i) Trimming the trees to prevent damage to a new fence
(ii) Sharing the cost of the new fence

4) If the neighbors are not forthcoming about sharing the fence cost, talk to the City or Municipal Planning Dept and see if they will step in to assist, you will be surprised some actually will. Also ask the City Planners what can be done about trimming the trees if the neighbors do not want to trim the trees, has to be phrased in the context that it is a hazard for property damage and personal safety (low lying branches that could fall)

5) If the City cannot help, I would go ahead and just rebuild the fence at the MIL's cost and trim the tree up to the fence, that way it cannot fall on the new fence. Some time it is not worth the frustration to deal with jacka$$ neighbors and you have to just dig in and do it.

The neighbor's might get vocal and rude about trimming "their" trees, but if it is over the property line into your MIL's yard, too bad so sad...

Good luck, let us know how it goes! By the way from that picture, it appears the entire fence in the back yard will need replacing.
Yasin