Lots of comments about iterating over children being O(N) for this style of trees. It's actually easy to generalize the atree design by e.g. adding pointers for "first child" and "next sibling" and potentially removing the parent pointers, if that's what you need in your application. I think the "Operations in psuedocode" section should simply state that there's no O(1) way to access children of a node - instead of recommending the O(N) approach, you should recommend changing the data structure to support the operations you need.

Storing nodes in arrays and using indices for pointers is a must whenever you're implementing algorithms on trees. I typically prefer using an array of structs, putting the key and the parent index next to each other, instead of putting them in separate arrays. If you need to access the children of a node, then be sure to consider if you can save memory by having a separate structure for leaves - remember that over half of the nodes will be leaves, so using space to store a child pointer on both internal nodes and leaves can be wasteful use of memory.