You compete on something other than price. If you really aren't offering anything different from what he is, and he is cheaper then you must expect not to have any business. Thankfully you have a lot more to offer. You just need to market that.
First, realise that hiring a tree service is for most people a once in a lifetime thing. They've got no idea how it works or what it costs, so they fall easy prey to guys with lies. They probably don't even have a friend who's hired a tree service. So take a few minutes to give them the real scoop, not just a price when you quote. I spend about 15 minutes on average at a customers house. Usual drill of being clean cut and wearing a uniform, being prompt and polite. I give them some info on the trees they have, which are good and bad and what they can expect from them. I upsell quite a bit of work this way. I don't give away too much, just enough to showcase my skills. I hilight the things about the job that are difficult or make it more expensive eg "that tree goes out over your house and your roof is made of tile which is old and brittle. Any small stub that hits it will go right through it. Luckily our guys are fully trained and we are familiar with this type of work. I'm going to charge you $1,700 for this tree to allow enough time to carefully lower off the branches and not do damage to your house." etc....
I also carry my company registration, public liability and workers comp as well as my arborist cert. It does make an impression. I make a point of telling the customer that a lot of guys claim to have it but if you ask to see it they wont call you back. Unfortunately a lot of hacks are now making fake certs!
I often throw in something a little extra rather than drop the price. This closes a lot of deals for me. Like if I'm doing a removal and notice they've got a cypress that's going to get big fast, I suggest it might be a good time to hedge it up and keep it under control. Adding something extra means that you're quote doesn't directly compare with the other guys. I also offer to supply free poison if they're not doing the stump. There's a million little things you can do to show your professionalism, and they all add up to your brand value.
Recognise too, that you can't and don't want to win all customers. Some folks plain don't have the money, and some just don't give a crap. Knowing your target market and target service will help you earn more.
For me, I stay away from the lower end of the market for obvious reasons, and also the upper end because I find rich people are a pain. They often expect too much and are difficult to deal with. Some guys specialise in that sort of stuff, and offer white glove treatment - no damage to lawns, manicured gardens etc.
Shaun