If it had the overall look of the 'trees in red' that are out all around the coasts in Wellington at the moment you could put it down as a quirky Metrosideros (pohutukawa) perhaps.

Or, if the gum trees are out in the Botanic Gardens - see if the foliage matches. (A Eucalyptus)

And - if you take your photo into the info kiosk in the Bot Gardens there are some very wise folk there who will probably recognise it instanter.

Thank you for the tips! Nevertheless I doubt whether it might be a red eucalyptus because the foliage does not seem to me quite similar.
The shape looks similar to the metrosideros, but the orange types are not as orange as this tree (if my Palmer manual & my book of native plants show the colours right).
I was also told about spathodea campanulata (in French 'tulip tree from Gabon') but I wonder if it would grow even in Coromandel!

Thank you for telling me I could ask at the botanic gardens, I did not know they had an information desk!!

It looks rather like Eucalyptus ficifolia to me. The flower colour is a bit variable but most of the trees I have seen are a bit more orange than pohutukawa and it flowers a little bit later. After the flowers there are seed capsules which are about an inch across - very distinctive.

From what I can see of the leaves and bark, these also look like they could be E.ficifolia.

It is a fantastic tree (I'm very fond of the Eucalypts, even though I can only identify a few species).

I think Spathodea campanulata is cultivated in a few places in New Zealand but I have never seen it and I think it would be diffcult to grow here. The individual flowers are quite big - you can kind of see where it gets the "tulip" part of its name when you see the flowers.