Integer binary tree with depth and max

Implement a binary tree of integers. How should an empty tree be represented? Write functions returning the tree's depth and its largest element.

Nodes hold a value and two child pointers; the empty tree is a null pointer. Depth is recursive: $0$ for empty, otherwise $1+\max$ of the children's depths. The maximum is the larger of the node value and the children's maxima, but it is undefined for an empty tree, so throw, return an optional or require non-empty input. Both are $O(n)$ time and $O(h)$ stack; manage node memory with smart pointers.