First this input :
// inserting elements
for (int i = 1; i <= 10; i++)
{
exampleTree.insert(i);
}
is the worst case scenario for a binary search tree, it will be equivalent to a linked list. So perhaps you should try some predetermined values so you know what the output should be.
Second, post order does what its supposed to. It prints in reverse order because your binary tree is like this :
[1]
/ \
[2]
/ \
[3]
/ \
...
since there is no left node, it just keeps going right, until it hits [10] then it prints 10, then goes back to [9] and prints 9 and so on.
Third , size, height, and depth are equal because of the tree structure.
Try to do manual inputs.