An important property of money is fungibility, which means each unit is equivalent to any other. When you break up a chain into sidechains, subchains, "subchains", treechains, mini-blockchains or whatever, you end up introducing differences among different units of the currency. That's the goal with colored coins, but less obvious is that fungibility is damaged even when you aren't marking the coins explicitly.
Those differences carry a different market value, and so they would by traded as different commodities, which is at least one reason it doesn't work very well as a scaling solution.
Put differently, coins on a sidechain, subchain, "subchain", mini-blockchain, treechain, or whatever represent a different financial asset than coins on Bitcoin proper, even if redeemable in bitcoins.
If, on the other hand, you design a system in which each chain really is equivalently secure on the blockchain, and protected from double spends, then you've really just obfuscated a scheme of faster, or bigger, blocks, one way or another, and the standard caveats concerning that applies. That appeared to be what Peter R's "subchains" did, but what do I know? Maybe I misunderstood him.
https://bitcointalk.org/index.php?topic=1320095.msg13492452#msg13492452There are various ways people have tried to get around this issue, but I haven't seen any that are all that compelling. For instance, Peter Todd says that in TreeChain's you can say that a transaction is safe once the work done on it exceeds the value of the transaction. At the end of the day it's indisputable that not all branches were created equal, and you're assuming different levels of security depending on where your transactions end up on the tree, thereby damaging fungibility. In the real world this would look like; "Oh you want to sell me a bitcoin? That's cool, but I'd like to wait for your branch to get more work on it before I feel comfortable doing that."
So what I'm trying to get at here is if you can somehow break up Bitcoin's transaction history while also preserving it's fungibility, that would be pretty neat.