Okay, here are some additional images: 1: underside of a leaf, 2: some early Autumnal leaves?, plus I pulled out my macro lens for a 3: closer shot of a "less decomposed spent reproductive part" (which this time had attached leaves so I am certain they are from the same tree) and 4: a closeup of a new leaf bud (terrible white balance but presumably still okay info-wise).
I am in Adelaide (Southern Australia - temperate climate).
Hope some of this helps, cheers.
Oh, and thank you John for all the info you posted there - it would be nice to be able to determine exactly what it is.