Falling Trees and Frivolous Lawsuits

This past July at 8:15AM a man decided to take a walk in Central Park in New York City.  After entering the park a branch from a tree fell off the tree and hit him.  The man was seriously injured and remains in the hospital to this day recouperating from his injuries.  His wife and him are suing the City of New York for negligency.  They claim the city should have known the dead branch was going to fall and should have removed it. 
My house sits close to the highway in woods that include dozens of Oak, Pine, and Cedar trees.  This year was especially tough on the tall trees in my yard and many branches have fallen.  This past summer I also fell from a branch knob from one of the trees.  The branch didn't hit me on the way down, but I didn't see it while walking, fell, and hit my knees on the sidewalk with lots of messy blood and pain.
As much as I try to keep my yard clear of such fallen branches from the trees I have wondered what would happen if a visitor of mine, or even a pizza delivery man, had an accident similar to mine.  I guess the home owners policy would cover it, but what if it didn't?  I could be left with a bill for a pretty penny to pay for medical services and pain and suffering.
Should the visitor to Central Park prevail in his lawsuit?  Or is it just one of those things that happens; an accident?  If he and his wife do prevail in their lawsuit what will happen with parks and open spaces maintained by cities as places of exercise, rest, or relaxation?
In the story in The New York Times, Kerry says

"Certainly this is a tragic case. However, nature happens. Park staff cannot possibly monitor each tree limb on each tree in the entire park every day. If this case wins in court, then the future freedom we experience in parks will be compromised. There will surely be signs everywhere to “enter at your own risk” and perhaps even waivers introduced that visitors must sign before enjoying the nature an open space has to offer. I understand that Mr. Blair-Goldensohn and his family must be devastated and looking for some type of payback for the damage this accident has caused in his life. But sometimes life is unfair, and there is no clear person or entity at fault, and we must live with the hand we are dealt, however bad that hand may be. I don’t think people who sue understand what precedent their case may create and the far-reaching consequences it may have."