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Is each block found just a link in the same long chain, or are there multiple chains?

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The Bitcoin blockchain converges to a single-file chain because each block references exactly one predecessor, and thus there can only be one block at every height in the chain connecting the genesis block and the chain-tip.

However, at times more than one block will be found at the same height. This happens either due to the mere happenstance of two miners discovering a valid block in the same instant, which is benign and will rectify by itself when one of the two sires a successor block and thus creates a new longest valid chain, or it could happen if a faction in Bitcoin adopts an incompatible protocol rule change (hardfork). Since network participants that don't accept the rule change will not accept these blocks as valid, two chain-tips may persist henceforth.

Recently, the hardfork plans of Bitcoin Unlimited have been a major topic in Bitcoin, which frequently includes discussion of possible scenarios with two incompatible chain-tips. I assume that you seeing a discussion of that sort gave you the idea that there could be more than one chain.

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