if the tree is in danger of falling down or is dead then my views are why quibble on who is supposed to do it. the tree can fall in either direction and cause damages to both sides depending upon where it lands. (Im assuming if it creates a big hole or uproots grass and such or hits buildings on your side, your side isnt going to leave that gaping hole or damages for others to get hurt on/ in. )

my point here where I am we dont fight over who's responsiblity it is to get rid of a dead tree. it doesnt really matter who gets rid of it as long as its done before it comes down harming others. if where we are we fought about dead trees and who's responsible for removing it the fight would go on until one side or the other no longer has a home to live in, pot to... well you get the point... with all the troubles in the world a dead tree removal vs paying for damages if it comes down on its own...

my suggestion would be to say to heck with this fight that is going no where and call the removers and have it removed. problem solved and both sides can get back to enjoying life rather than fighting...

but since you asked who would be responsible... here the city code is if a fence is involved the responsible party is the one who resides on the side of the fence where the tree is..... unless.... the tree is on city property (with in a certain distance of roads and sidewalks) then its the responsibility of the city.

my suggestion contact the city. they will tell you whether you or they are responsible according to your locations city/ town codes and if it is a danger or dead the city will contact you or your neighbor with a violation of coding correction request that will require you or your neighbor to remove the tree with in so many days.