This is one of those questions that everyone seems to write about but the same bad explanations are just plagiarized from site to site.
Edit: The heart of my question is why Merkle trees are claimed to be efficient when determining if a leaf node is present or tampered with when only the root node hash and a leaf node hash value is known and when the Merkle hashes are not cached somewhere (which they appear not to be in the case of Bitcoin's transactions stored in a block)... I so far see only claims of efficiency not any credible demonstrations or explanations of same. The whole point, I understand, of the Merkle "proof" is that the hashes are stored and then the binary feature of the Merkle tree can be leveraged to "walk the tree" from leaf to root with only a fraction of hash calculations required; without the cached hashes, I submit that all hashes in the entire tree would need to be recalculated, as I explain below...
I understand how to build Merkle trees. What I don't understand are two key points about Merkle trees and one key point about Bitcoin's Merkle trees for transactions. For the sake of my questions below, consider this cited Merkle tree from investopedia.com:
Merkle Trees in General
People say that given the Merkle root hash and the hash to H(D), you can quickly determine:
A. Whether the H(D) leaf is in the Merkle tree.
B. Whether the H(D) leaf has been tampered with or whether it changed positions.
With respect to A: To do this, wouldn't you just look at the list of leaf hashes to determine whether H(D) is there? Why do anything with the Merkle root?
With respect to B: If you have the H(D) hash value and you know that its leaf pair is H(C), you can calculate H(CD). But at that point, if H(D) has changed and you know the old value for H(CD), wouldn't you immediately be able to short-circuit your investigation because the new H(CD) value would not equal the old H(CD) value? If you say to me that you don't have the old H(CD) value then I would ask you: How do I get the value to H(AB) then? It would seem that I need to recalculate the hash for H(AB). And if that's true, I also would need to recalculate the hash to H(EFGH), which means recalculating the entire right half of the Merkle tree starting at the leaf nodes. There isn't much efficiency in that.
My final question:
C. Does Bitcoin persist the Merkle tree hashes for not only the leaf nodes but for the entire tree all the way to the root? Where exactly and how exactly does it store this information? All I ever see is the Merkle root in the block... I think I once saw a Merkle value for a transaction node but I never saw any meta data related to non-tree Merkle nodes in the Bitcoin blockchain...
Thanks!