Nodes displayed on tiles

Nodes are not suitable for the upper levels of the hierarchy because you cannot find out which other nodes are "below" them. Nodes are only suitable for the lowest level of the hierarchy.

One possible way is to install your own copy of Nominatim and extract the pre-computed hierarchies from Nominatim either through the API or by querying the database directly.

What you can also do is use osm2pgsql to import OSM data for the region you are interested in into a local PostGIS database; you can modify the "style" file to only load "boundary" and "place" objects to speed up the process.

Then, you can build SQL queries that give you the hierarchy you want, for example to compute a list of cities and their suburbs

SELECT city.name, suburb.name 
FROM planet_osm_polygon city, planet_osm_point suburb
WHERE city.boundary='administrative' AND city.admin_level='8'
AND suburb.place='suburb'
AND st_contains(city.way, suburb.way);

And so on. (Note that this assumes cities are on admin level 8 which some larger cities might not be, and that suburbs are just points - they could also be polygons so you'd want to repeat the query above for polygonal suburbs, etc.)