an immediate consequence of the height-balance property is that subtree of an avl tree is itself an avl tree the height-balance property has also the important consequence of keeping the height smallas shown in the following proposition proposition the height of an avl tree storing entries is (log njustificationinstead of trying to find an upper bound on the height of an avl tree directlyit turns out to be easier to work on the "inverse problemof finding lower bound on the minimum number of nodes (hof an avl tree with height we will show that (hgrows at least exponentially from thisit will be an easy step to derive that the height of an avl tree storing entries is (log nwe begin by noting that ( and ( because an avl tree of height must have exactly one node and an avl tree of height must have at least two nodes nowan avl tree with the minimum number of nodes having height for > is such that both its subtrees are avl trees with the minimum number of nodesone with height and the other with height taking the root into accountwe obtain the following formula that relates (hto ( and ( )for > ( ( ( ( at this pointthe reader familiar with the properties of fibonacci progressions (section and exercise - will already see that (his function exponential in to formalize that observationwe proceed as follows formula implies that (his strictly increasing function of thuswe know that ( ( replacing ( with ( in formula and dropping the we getfor > ( ( ( formula indicates that (hat least doubles each time increases by which intuitively means that (hgrows exponentially to show this fact in formal waywe apply formula repeatedlyyielding the following series of inequalitiesn( ( ( ( ( ( that isn( * ( )for any integer isuch that > since we already know the values of ( and ( )we pick so that is equal to either or that iswe pick