There are a LOT of Posts on this Forum about drainage and French Drains. User your Browser's FIND (or SEARCH) Feature to search the page for the word drain. DON'T stop searching with the first 'hit' because I, among others, have responed to many posts on this.
Also, follow up your reading with this page. Intro To Landscape Drainage DON'T use Weed Block where it talks about Landscape Fabric. I prefer Filter Fabric.
You need to address your 'root' problem(s). If you don't, they'll be back. Move the trees or cut them down. If you're planting new trees, keep them small. The root systems usually extend about 3' BEYOND the tree's leaf-line. That will give you an idea of where your roots are. If a tree is 10' in DIAMETER, then the newest roots are 8' from the trunk. (That's 10' / 2, plus 3' = 8') Your new plumbing should NOT be inside the 8' 'window'. See what I mean???
As for your neighbor helping you out, they have no obligation UNLESS they created a water problem when there wasn't one in the past. If this IS the case, they can be sued. If the runoff is natural, it's your problem. Secondly, if YOU create a runoff problem for them, YOU can be sued. Make sure the water you're 'collecting' is drained to an area (or street) where it won't cause harm to other residents. If you need more specifics on anything, e-mail me directly.
There's a great article in the current issue of The Family Handyman Magazine that addresses this job at length. The issue is worth buying! It's the cover story. My best to ya and hope this helps.