Ok, I have some experience with mangrove roots. I am from Hawaii and they have been introduced there. I went out and collected some live mangroves (Rhizophora mangle) from brackish/full seawater environments, and here is what I found.
First of all, it is impossible to use the roots submerged unless you have a huge tree connected to them on top. This is because the roots are actually very buoyant on their own (something you can't see in pictures is that the whole tree would float if you ripped it out of the deep sediment because of their roots), so even if you could get a hold of them they would be impractical to use. Also, since they are not a hardwood but rather a very spongy softwood they rot immediately. A further problem is that big mangrove trees cannot adjust to freshwater from salt/brack. I took several rather large trees and none of them grew any new roots or leaves, and after a period of about a month all their leaves dropped off and the plants just died. Seedlings however do great.
Also, growing the seedlings into the 'shape you want' is totally impractical. First of all I was growing mangrove seedlings in ideal conditions. Full Hawaiian sun, running freshwater with lots of nutrients (out of the back of my tank and in HOB filters). I can say that in these conditions my plants took about a year to grow 18inches tall and develop ~12" long roots. They also had zero branches, they just grew straight up (you have to bonsai cut them to get branchy-ness, which I didn't have the heart to do since they grow so painfully slow). Let me reiterate are a VERY slow grower. The other problem is that they get infested with pests (aphids) that are impossible to eradicate.
Also the roots which came out of the seedling DO NOT look anything like the great aerial roots you see from trees. They are thin (like a pencil), are full of feather like hairs, and for the most part point straight down. If you try to prune them, bit just come up from the sides of the prune and go straight down again. So there is no way to control what these roots do.
So in summary, the only way to get real mangrove roots in your tank (Rhizophora Mangle at least) is if you had maybe 5+ years plus to grow the thing under halides and after that the roots will still look nothing like you planned.
The best you can do is get other woods that look like mangroves, or use artificial stuff.
Cheers