If somebody is thanking you, you likely exerted effort on their behalf. So they say “thank you” for exerting effort, and then you respond with exerting effort was “no problem” you don’t need to thank me. You’re welcome would be a response more along the lines of “you are welcome” to make me exert effort.
They’re both fine, and it’s both a generational and locational thing. I personally prefer “no problem” because honestly I don’t mind helping, but I’m also not giving you permission to use my effort at any time. You can have my efforts when it doesn’t cause a problem.
I’m from the US and “better than a kick in the teeth” and “better than a poke in the eye” are both common around my area. Never heard the tree ones though.