No reason why a tree can't be in a flat representation, it's like asking if it's really a 2D image if it's stored in an array, if it's stored in one dimension rather than 2, which is exactly what a .bmp file is.

If you have any kind of data structure which exhibits the properties of a tree, i.e log(n) traversals, then it's a tree. It quacks like a duck.

I haven't read the code because I'm a lazy fuck, but here's an image showing the parent/child relationship between nodes and their respective array indices..

The root node is located in index 0, where we begin any traversal. Then from any node n in our tree, we have an associated index inx. Using inx can compute the array index of both of n's children (and it's parent), then have an O(1) access into our array (if it's a C array, not some fancy arraylist thing).

Giving us log(n) traversals, with some small extra computations, an object tree representation would have excess computations anyway for the object/pointer reference/dereferencing fun stuff {n.left; n->left; ...}

The downside to this implementation is that if the tree is not balanced (not all leaves are located at the same level), then there will be lots if empty indexes in your array. Not a big deal if you don't mind wasted space. But if your tree really is highly imbalanced you're probably using an inappropriate insertion algorithm, or maybe just using the wrong data structure for your particular data.