If you have a drainage issue, or you're likely to have freezing conditions, I'd personally want to lay a small coating of gravel across the concrete first - allow that pent-up water to get out of the way.
If you're building this in, say, Phoenix, though, that may not really matter and laying them directly atop the concrete shouldn't be a problem.
To the 'cement' part of your question: why not go with dry-laid? As long as there is some form of rim to the patio, or you're not driving over them, dry-laying would make both maintenance and modification easier.
This is a far easier job than you may think.
First, pry out the brick. An old screwdriver will help. Stick it in one end to pry up, then the other end. Before long you will work it out. Once it is high enough, you can grab it with your hand, pulling straight up. You will want to remove a couple of bricks in that area, since there is a root to deal with. Of course, once the first brick is out, its neighbors will come out easily.
Once removed, deal with the root. Cut/shave it back with an old hatchet perhaps. (Yes, it will dull the blade, digging in the dirt. I did say to use an old hatchet.) And, don't worry, the parent tree will survive this bit of surgery.
Now, pack the area with stone dust. You can buy this by the bagful from your local home center for only a few dollars. (In a pinch, sand is ok, but it does not pack as well.) Bring this back to a perfect level. As you put a bit of the dust in, I'd tamp it down using the end of one of the bricks you just removed. You want this underlayment fully packed in, with no voids. Later settling is something you don't want. So I might lay down a half inch of dust, then tamp it in, then another layer of dust. Finally, smooth it out using a small trowel if you have one, or just the edge of one of those bricks. Get the level of the underlayment just right, so that when you replace the brick it will be perfect.
Now lay down the bricks in place again. A rubber mallet is good here as a tool of "persuasion", in case any are too high. Tap on all of the bricks to make sure they are settled in. If a brick is too low, pull it out, and add some more stone dust. You want the final surface to be as smooth as a babies bottom. Trip points are to be avoided here.
When the bricks are back in place, lay down some more of the stone dust on top, and sweep back and forth with a push broom. Work the dust into the cracks between the bricks. This locks them in place. It may require another pass with the push broom and more stone dust in a few days to fill in the cracks, or after the first good rain.
In all, expect this to be an easy job though, taking perhaps 30 minutes to do the complete repair.
Best Answer
I would take a 4-1/2" angle grinder with a diamond wheel and cut out the mortar between the bricks down level with the patio surface, at like 5-10 brick intervals, creating a path for the water to flow out. Then re-caulk the joint tapering the caulk at each "drain".