The time complexity of merge (union) operation is said to be $O(\lg (n_1 + n_2))$, where $n_1$ and $n_2$ are the numbers of elements in the merged heaps, respectively. I do not understand this - the algorithm has to go through all the elements of both rightmost paths of the original heaps - lengths of these paths are bound by $O(\lg n_1)$ and $O(\lg n_2)$. That makes $O(\lg n_1 + \lg n_2)$ in total, which is $O(\lg (n_1 n_2))$. Where am I making a mistake in my assumptions?
Arbitrary delete operation - the complexity should be $O(\lg n)$, where $n$ is the size of the heap. But after the deletion, the algorithm has to go through all the nodes from the parent of the deleted node to the root and correct the Leftist property, and the lenght of this path is bound by $O(n)$. Again, where am I wrong?