You can eat the fruits of the Lychee too, if I'm thinking of the some plant. They are white fruits, right? They are canned and you can get them in Asian grocery stores and sometimes regular stores. Those and Longons are on chinese buffets. I picked Longons and took them home when I was in Florida a few years ago. They are great fresh, but, don't keep well on a 19 hour drive back to Pa from Florida. I wanted my dad to taste them fresh, not just canned. We cut the bad parts off, I can see why they aren't in stores like fresh apricots and peaches etc. Lychees have a nice delicate flavor. I never had a fresh one, just canned too.

Not enough to go on - esp. the pieces that have been dead for some time

bugs have a way of finding stressed plants .... and a tree that caught fire is probably stressed .... 3 years isnt much time in tree time.. which is counted in decades ... probably one of the wooly aphid type creatures .. but i cant tell from the pix ... but it looks awful close to the house... it might be worth your while to have a certified arborist give you an onsite inspection of the trees overall health ... add your big city name to the link below ken https://duckduckgo.com/?t=ffab&q=isa+certified+arborist&ia=web

find some leaves on the ground.. and take a pic of those with a scale.. if you can ... reminds me a bit of redbud or poplar with those heart shaped leaves ... but they are too small in these pix to tell for sure ... whether or not something might fall down.. really depends on whether it will fall on anything ... of value or importance .. this tree looks like it pretty far from anything important.. so why would there be any imperative to have it removed.. btw.. i dont recommend chopping it.. unless you just want some backbreaking aerobic exercise .. lol ... and i presume the car isnt parked there all the time .... in theory.. any tree might fall down.. but we dont really go bare earth to avoid such.. its all about the odds ... and starting with an ID was spot on .... anyway... in tree time.. counted in decades.. ya.. it might fall down.. i dont see any reason why it need be done inside a few weeks or months ... ken ps: perhaps your realtor ought to stick to selling houses.. i dont see any indication from the lawn.. that the tree is aggressive with stealing water .. if it were.. grass wouldnt grow there.. and i note.. there are plants growing under it.. right at the trunk... that doesnt happen with water hog trees .... and btw.. if you decide to have it removed.. thats a bargaining point for a low offer.. if the market will bare such ... on reread.. i see you already bought the house.. congrats .. pps: whats a neighborhood gardener.. and why does he or she have a hate on for this tree ... probably his car parked under it... lol ... or he wants you to hire him to remove it????