Claim for when the initial branch fell and use that as leverage to get it removed?

I bought my house in 2008. In July 2009, I had a large limb fall from a tree near my house. It landed on my deck, smashed a patio table, and dented an eave trough. I repaired the damage myself and did not file a claim with my insurance company (In hindsight, I should have filed).

Fast forward to summer 2013, the tree has sprouted mushrooms: http://imgur.com/wcn8sov
Where there are mushrooms there is dead wood. The large branch going off to the left in the picture extends over my garage and my neighbor's house. If it fell it would cause serious damage to both our houses.

I want the tree removed. I had an estimate by a local company and they said it would cost $5000 (!!!). Probably because it extends over 2 houses and there is not a great way to get a large vehicle into my backyard. I cannot afford $5000 for tree removal.
So there it is. What should I do? Can I file a backdated claim for when the initial branch fell and use that as leverage to get it removed? Or, will they just note it on my file and tell me it is my responsibility since it is a known hazard? My response to that is I legitimately did not know it was a hazard before the first branch fell in 2009.

Your window is closed on that particular claim, you only have 2 years to file. Some policies do cover falling objects some do not. Tree removal is also not always covered and in my state is capped at a fixed amount $250 per tree and it already has to be resting on a covered peice of property.

If they were going to provide coverage you would still be subject to your deductible.
Honestly that price for tree removal sounds insane I have had very large trees removed and stumps ground down for $1000. I think the best course of action is to shop around. I would start the conversastion with "I have a company offering to take it out and grind the stump for $1000 cash but I wanted to see if I could get a better price".

Most insurance carriers will not cover removal of standing trees. It's considered normal maintenance that the homeowner should cover.
You could wait for it to fall, when your insurer would kick in, but that is a risk proposition. It could cause massive damage or injure someone in a way that would exceed your insurance limits. Also, most carriers do not include high limits for tree removal if it falls (usually only $500), so you'll probably still get stuck with most of the bill, even if it doesn't damage anything.
Sorry to be the bearer of bad news, but I would get more tree removal quotes and bite the bullet. It's the joy of home ownership.

I'm going to have to agree with them both and say you're going to have to foot this removal yourself. Normally though when a tree falls and hits your neighbors home, their insurance would cover their damage BUT since this tree has been a problem already you'd probably be held liable for any damage to their home also.
Seriously get more quotes and see about maybe just starting with that large, dangerous part?

You should also speak with your neighbor, show them the tree, and ask them pay for 1/2 or whatever percentage of the tree would cover/damage their house if it falls as well.
Then again, that may be a dangerous ground because if they say no and you don't do take care of the tree, and then it falls on their house, couldn't they then sue you or go after your insurance company??

You cannot file a backdated claim from 2008. Statute of limitations I believe is 2 years. Not to sound rude, but this tree is on your property and it is your responsibility to maintain it. You said above you realize the tree is dead, so it your responsibility to mitigate any further damages by trimming any branches over your neighbor and/or your property, or really, have it removed! You didn't know it was a hazard in 2009 which is OK as far as the damage incurred then, but you've known for 4 years since then and haven't done anything about it. It is your responsibility to either pay for its removal or pay for any damages your or your neighbor incurs from any future damage. Also, your neighbor has the right to cut off any growth (branches) that enter their yard from your tree. They can go from the property line and cut off the branch. However, they might not be aware the tree is dead as you say, and it is also on your property, so that really lies with you, since you technically own the tree and you are aware of its condition.
I worked as a property claims adjuster for 2 years. Tree removal is expensive but $5000 sounds expensive. I would shop around and then get it taken care of ASAP.