You don't need to cd to directory to use tree command on it, and you can specify multiple directories to use tree on them at the same time.
tree /dir1 /dir2 /dir3 -o /some/other/path/outputfile -n
also if you want to use stdout redirection like > or >> you can specify a path there also
tree /dir1 /dir2 /dir3 -n > /some/other/path/outputfile