Merkle trees are everywhere in Bitcoin (full node or otherwise).
The block header has the merkle root of all transactions that are a part of that block. The witness merkle root is stored in the coinbase transaction (if the miner is segwit enabled).
And proof of work is done for the block header, which includes all these merkle roots.
The block header has the merkle root of all transactions that are a part of that block. The witness merkle root is stored in the coinbase transaction (if the miner is segwit enabled).
And proof of work is done for the block header, which includes all these merkle roots.