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Merkle tree recalculation during transaction lookup

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When full-node receives a request from SPV client to check if given transaction exists in the blockchain - does it always re-calculate the merkle tree on the fly in order to return merkle path + block header to SPV client for given transaction ID?

How does it look like from performance point of view? Is there is any merkle tree caching for example? Because as I understand the node will have to do the following steps:

  1. Find a block in which given transaction ID exists
  2. Fetch all transactions from that block
  3. Calculate merkle tree
  4. Return merkle path with relevant hashes necessary to confirm merkle root

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