Calculating the root of the commitment tree is not useful. It is public information that can be obtained by anyone. It is even part of the block header.
We did it for illustration purpose and to get familiar with the problem space.
Now, let's turn to the real problem: Merkle Proofs.
Quick Reminder
The Merkle Proof for a note N at position p is the triplet (N, p, $\pi$) where $\pi$ is the Merkle Path: The list of the siblings (hashes) from N to the root.
Let's look our example tree:
The Merkle Proof for the note L is (L, 4, [M, G, B]). It enables the verifier to compute the root A.
The position 4 gives the path to the note:
 write p is binary. p = 100b;
 replace 1 by R and 0 by L. The path is R, L, L;
 reverse it to get the path from the note to the root: L, L, R;
 Use the sibling hashes to compute the intermediate nodes.
We start with x := L^{1}.
 First step is Left and the sibling is M: compute x := Hash(x, M);
 Second step is Left again. Compute x := Hash(x, G);
 Last step is Right. Compute x := Hash(B, x).
At this point, we have x = A.
It is not computationally feasible to get another proof for L because of the properties of the Hash Function. Neither can we make a proof for a note that does not belong to the commitment tree.
There are lots of notes in the tree but we own very few of them: usually fewer than 1/1 million.
The goal is to produce the Merkle Proof for the notes we own (that we didn't spent) whenever we want to spend, i.e. at the current block height.
Again, we'll start with a simplified problem.
What state should we keep with our notes?
Footnotes
Footnotes

The sign := stands for "assign the variable on the left hand side to expression on the right hand side". ↩