A palm tree that grows to about 10m high?

nothana
Nikau palm tree is another New Zealand native tree which fits your description. Have a look at these websites:

http://www.peersbrownmiller.co.nz/some_native_trees_2/some_native_trees_2.php?mid=258

http://www.conservation.net.nz/upload/documents/getting-involved/students-and-teachers/field-trips-by-region/kapiti-education-kit/Species%20cards/nikau.pdf

http://www.Wordworx.co.nz/TitirangiNikau.htm