Consider a binary, perfectly balanced tree of processes of height n where all leaf nodes have the same distance from the root; so there is a total number of m = 2 ^{n − 1} nodes in the tree. Assume that the root is the coordinator of the commit protocol. Determine the number of messages and forced log writes for the presumed-nothing (i.e., basic 2PC), presumed-abort, and presumed-commit protocols for the following situations:

(a) all processes have performed updates and the transaction commits,

(b) all processes have performed updates and the transaction aborts,

(c) all nonroot nodes are read-only and the transaction commits,

(d) all leaf nodes are read-only and the transaction commits