I think you mean "up A tree." It means that you are cornered, and have no way out of a situation. The image is from hunting, when a dog would corner an animal up a tree. The animal could not come down, but it could not escape, either. "Up a tree" means you have no solution in sight.
It means that she is crazy- mad like a dog.
It's not an idiom, it's a joke. And it's "make like a tree and leaf" -- it's a pun.
No, it is an idiom because it doesn't seem to mean anything unless you know the meaning.
An idiom
A mistake or false
It means things are trustworthy.
To make a mistake
About to be sold, or given up.
say no to it
The idiom "barking up the wrong tree" originated from hunting dogs that would bark at the wrong tree instead of the one where the prey was hiding. It means pursuing a mistaken or misguided course of action, like accusing the wrong person or looking in the wrong place for an answer.
It means shut up!
flatter her to get something