The longest path from the root node of an AVL tree to its most distant leaf consists of five nodes, including that root and that leaf node. $($There are three nodes in the path between the root node and its most distant leaf.$)$ What is the minimum number of nodes must the entire AVL$-$tree have to have to be able to qualify as a legal AVL$-$tree? $($Start with a tree that is a linear branch of five nodes, and add all nodes that must necessarily exist for this tree to satisfy the AVL property.$)$