Issue One Inconsistent Notations: public/private parent/child key and parent/child public/private key
The above two kinds of notations are used around the BIP32 document, while I suggest it is better to uniform the whole context with the 2nd kind of notation for better clarity and reasoning. Since in a public key cryptosystem, the key pair contains a public key and a private key, and here all key pairs are associated with either parent nodes or child nodes, it makes much more sense to say parent public key than public parent key.
Thus, for example, the subsection title, “Public parent key -> public child key”, should be “Parent public key -> child public key”. Another subtitle title, “Public parent key -> private child key”, should be “Parent public key -> child private key”.
Issue Two Improper (Incorrect) subsection title: “Private parent key -> public child key”
The function N((k,c)) -> (K,c) is simply the computation K = k * G. It computes the child public key from a child private key or the parent public key from a parent private key.
Thus, the subsection title “Private parent key -> public child key” should be changed to “Private key -> public key”.