Sorry that happened. I've never seen a real slate roof. I guess I'd start asking around for contractor recommendations.

You need a roofer who is VERY experienced repairing slate. It can be repaired but it's not like putting asphalt shingles up - you need an expert. And the damage should be paid for by the tree removal company - it's their carelessness that caused this.

You need a pro roofer with experience in fixing real slate roofs. They are definitely fixable, but it's a talent.

What they will do, depending on where the damage is and how extensive, is remove individual damaged slates by reaching up under it with a special tool and removing the nails holding it. Then they insert special holding tabs, slide in the new slate and bend the tabs up to hold it. Tabs used only in no-snow areas, there is a special "slate hook" or you nail the slate near the top and put a wee bit of flashing over the nail to prevent leaks.

If the damage is widespread, they carefully remove a patch of slates, toss the broken ones, and re-do the patch, adding new slates as needed.

And the tree cutters should have insurance to cover it.

https://www.nps.gov/tps/how-to-preserve/briefs/29-slate-roofs.htm 

The application of roofing mastic or sealants to damaged slates should
not be considered a viable repair alternative because these materials,
though effective at first, will eventually harden and crack, thereby
allowing water to enter.

http://www.slateroofcentral.com/repair_slate_roof.html