A* Search, node to be expanded next when evaluation function evaluates the same

Question

Im having trouble understanding which node/state should be expanded next in an A* search tree when the evaluation function (f(n) = g(n) + h(n)) evaluates the same for two nodes.
Example 1

my understanding is that the frontier is stored as a priority queue ordered by f and therefore as the nodes on the frontier have the same value the the node added to the queue first will be evaluated.

Example 2

In this example the evaluation function of B was less than C and therefore expanded but generated a node D with the same f as C, in this case what node would be chosen for expansion next?

(I realise this question should probably have been posted on cstheory.stackexchange but I don't have enough reputation to post images, apologies)
Thanks in advance

Answer 1

Doesn't matter which one will be chosen, depends on priority queue implementation, but more often it will be C, because newly created node D won't went before C which is already in queue. If we continue with C and later on we realize that h(C) was underestimated, than the f value of current node (accessor of C) becames greater than f(D) than the algorithm will go back and expand D as it become top of queue. This will work since heuristic h should always gave smaller or equal values than the real cost.