Man Trimmed My Tree Without Any Written Consent And Is Threatening A Lien On My Home If I Don't Pay Him

Title is about as dumb as it sounds, and it's a long story so please bare with me!

I have a crazy neighbor (we'll call her Pam). I had gotten a quote from a professional company to trim a very large tree in my front yard that sits nearby the property line with my neighbor. The quote was higher than I anticipated and I realized I wasn't in the market until I saved up for a bit. Pam overheard this and in the following weeks has sent multiple people to my door to give me low offers on trimming my tree. The first two were unlicensed and just broke guys offering to do it for some quick cash, to which I immediately said no to.

The third guy (we'll call him Tom) gave me a decent offer and showed that he was licensed, however I told him that I was not in the market at the moment due to some unforeseen bills to the vet. Tom says he's going to be trimming the neighbor's tree and will be around, doesn't want to "leave me hanging", and will probably fix it up a little bit anyways. I basically say that if he wants to, go right ahead, but I'm not in the market to pay.

A few days later, Tom and his team are trimming the neighbor's tree in their backyard, and I come home to see they are walking down the side of my yard. I find out they have destroyed a part of my fence in order to remove the logs and debris that fell in to my yard, WITHOUT my permission. They say they will replace it (they do replace it btw), and Tom starts trying to tell me how he wants to trim my tree. Pam comes out and says that she's wanting the tree trimmed. I ask Tom if Pam is footing the bill, because last I said is that I cannot pay for it now. I shortly entertain a payment plan but no hard numbers were ever agreed on. He says "We'll figure it out", and that's about the end to the conversation outside of small talk.

The next day, they are trimming my tree. I didn't say anything to them, but assumed that maybe Pam decided to foot the bill because nothing was hard agreed on, and I never signed anything. The day after they are done, sure enough Tom shows up and hands me an invoice for the full amount that he originally quoted me that I said no to. It says "make payment plan with customer". I say "uhh did I ever agree to this?" and he's stumbling words to basically not answer. I'm pretty mad now and explain that I assumed somebody else was paying because I never signed anything or agreed to a specific price. Still stumbling, he basically comes to say that I can pay him what I want because he doesn't want me to be mad at him or his company.

Now for the fun part. It's been two weeks, and I haven't heard anything from him. I'd been considering throwing a few hundred at him since he did do a lot of work, however I come home one day to find his business card in my door. It reads:

"I will resolve your bills for $600 today, if not I will have to file a lien on your home. Let me know."

I have been laughing thinking about it every day since, but it does make me wonder, is there anything he can actually do? Is there anything that I should do? I wanted to contact him and tell him no but I don't want him to have my number. I know I should have stopped them when I noticed them doing the work but I assumed that it was worked out with the neighbor since I NEVER signed anything or agreed to a firm price. Any advice is much appreciated!! I am in Oklahoma if that is relevant.

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TLDR: guy offers quote to trim tree, I say no. He is persistent and I consider, but NEVER signed any contract or agreed on set price. He trims tree, sends me invoice for full original quote I said no to, and two weeks later leaves message that I need to pay or he will put a lien on my home.