A* Search?

In my knowledge,

Greedy is using f(n) = h(n), where h(n) is the heuristic function

A* is using f(n) = g(n) + h(n), where g(n) is the accumulate-cost

function

This is the only difference between 2

I find a statement in a book:

"Greedy is only compare the current expanded node's children, and A*

is compare all left nodes"

Could anyone tell me is this statement correct??