Does Merkle Tree root hash changed when miner node not found POW nonce?

Does Merkle Tree root hash changed when miner node not found POW nonce?

In bitcoin PoW process,if miner does not found the nonce, I understand that he must to increase nonce and rehash the entire block header again to mine the new block. so if nonce defined 4byte[between 0 and 2^31], goes from 0 to 2^31 finally and even not found the target hash, does the miner rehash the Merkle Tree including coinbase transaction? And what algorithm for miner to change coinbase transaction?

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