Depending on the damage and your deductible, might be worth it to make a claim. Document your labor time too and take lots of pics, as that and the tree removal and the roof repairs should be covered. 

If you can’t get someone there asap for the removal and you do it yourself, you might want to grab some tarps, screws and some 1x4 strapping to beat any incoming precipitation and protect your attic insulation and textured ceilings in the living space. Just make sure you lap your tarps up over the peak to prevent any rain from running underneath them. 

I’ve done a bunch of these claims in my days as a restoration project manager. And now do the residential and commercial estimating for a good sized tree company. 

Judging by the bark and the leaves in the picture it looks like it’s a variety of Willow. They are quite susceptible to internal decay because water gets into the bark conclusions in some of the crotches. Often there’s a little to no outward sign of this .