So half a bit would be like hearing "Umm..." in response to that same question?
"Is this tree deciduous?" "Yes" - one bit
"Is this tree evergreen?" "Probably" - one bit (they don't know), another bit (their guess)
But I'm way off any formal understanding of this, and it can be rigorously defined.
Meta comment: "half a bit" was clearly a joke, and so was my response; now I'm taking your reply at face value and debating it seriously, while admitting that I haven't got anything close to firm enough ground under me to actually debate it :)
"Umm..." definitely carries some kind of information, but it doesn't actually help answer the question.
Based on the information theoretic definition of entropy we'd need to go from 50:50 to 89% certainty to get half a bit of information, and I'd probably qualify 89% certainty as "probably"