Hi paallo90
As mentioned earlier in this thread, there are a couple of family trees at the ancestry.com website for the TRESEDER surname.
The father of Rena Tui TRESEDER is said to have been born in Australia, and the mother Ellen Jane MARTIN, in Southland, NZ - her parents names are not given.
--------------------
It doesn't necessarily follow though that just because the child was given a "Maori" name, that she had Maori ancestry. The Christian name "Tui" was a popular name for both NZ-born girls and boys, especially from around the 1890's ... "Huia" was another one. [I have a number of both these names in my various non-Maori families ].
The name "Rena" comes from a variety of cultures. [The usual spelling of the Maori variant appears to be "Rina" ].