But seriously - what need does a tree have for money? or hair??
If going so far as to give an offering in return, we should think of what has value to the tree, not ourselves.
It's not about the money (a penny would be totally inappropriate in that case: that's an insulting offering if we're talking about money as money). I think it's mostly about having a simple physical object you can anchor the energetic offering to. Just dumping it on the tree seems rude, the same way that dumping it on someone else would be: by transferring the offered energy to a secondary object nearby, it's like sticking it out on a plate: the tree can use it or not.
A penny or hair is used because they're common objects, and either biodegradable or not going to cause problems to the local environment (pennies will eventually leach generally beneficial minerals, hair will degrade or end up in a bird's nest.) Same deal with my handspun undyed yarn: it contains my energy, but it's not going to introduce modern dyes to the ecosystem or anything.
If you can't find a wind blown branch, and must cut, personally, I feel it'd be better to water the tree and fertilize it, regularly, if possible, over a week or two; also, apply some kind of apropos "salve" to the end of the cut branch, though I don't know yet what would work best other than dark paint. (I need some research there) ..something though, to keep bugs out, and moisture in, and make sure you don't endanger the health of the tree.
My understanding is that sealing the tree is actually *harmful* to a number of species - either because it's not what the tree needs, or because a particular species has a reaction to a common ingredient. (I am not informed enough to know which trees, however - in the US, this is something local agricultural services would likely be able to tell you.)
As far as watering/fertilising - I don't know about anyone else, but many of my own wood items have come from public land (or something like a cemetery, like the yew wand my HPS cut for me at one point) which is already either being tended, or designed to be *untended*. In the first case, additional things like fertiliser might either affect the local water run-off, or be in contradiction to the tree's needs, or whatever.
Water, however, when the weather's dry, would be good. But again, probably not actually helpful except as a salve to the conscience if you're in normal weather patterns for the area.