Simply withdraw to teatree from Bittrex. Memo field is optional for withdraws from Bittrex (deposits to Bittrex are mandatory so they know who sent it though). Registered account just means your account. They can't send to an account that doesn't exist hence "registered."
If the purpose of buying steem is to increase your SteemPower, go to your wallet page after confirming the withdraw, click on your STEEM balance, and select Power Up. The sooner you do this the better, as the steem supply always inflates but your steem value will stay static when it's put into SteemPower.
But which address is the registered one? On the "permissions" tab of my account I have four addresses - posting, active, owner and memo. Is "owner" the registered address?
Those are not your addresses. Those are access keys (similar to passwords). As arhag said, your address is just your account name 'teatree'
Transfers in Steem use account names not addresses (unlike Bitcoin and many other cryptocurrencies). So what Bittrex is looking for is the human-readable account name (e.g. teatree).