Actually Carica papaya is a tree by any definition you care to select, which
cannot be said for banana 'trees',

Why? As far as I could tell from the pictures, they are both herbaceous. The
references on the Internet call them plants, not trees.

palms and bonsai.

What is the trunk of a palm made of? If it is lignified in some way, then I
would call it a tree. Does the absence of branches make it not a tree?

Why is a bonsai not a tree? It may be only a few inches or a couple of feet
tall, but
1. In most cases, if it were put in the ground and allowed to grow freely, it
would become a normal size tree.
2. Its purpose is to represent a full size tree. So why isn't a bonsai a tree?

Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)