# Data Structures: Question Set – 24

#### What is the difference between a height-balanced tree and an AVL tree?

A height-balanced tree is any binary search tree in which the heights of the left and right subtrees of any node differ by at most a constant factor (typically 2). An AVL tree is a specific type of height-balanced tree in which the factor is always 1.

#### What is the minimum number of nodes in an AVL tree of height h?

The minimum number of nodes in an AVL tree of height h is given by the recurrence relation N(h) = N(h-1) + N(h-2) + 1, with base cases N(0) = 1 and N(1) = 2, which corresponds to the Fibonacci sequence. Therefore, the minimum number of nodes in an AVL tree of height h is roughly proportional to the golden ratio raised to the power of h+1, or about 1.618^{(h+1)}.

#### Can an AVL tree be implemented with an array instead of pointers?

Yes, an AVL tree can be implemented with an array instead of pointers by using the array indices to represent the nodes of the tree and storing the left and right children of a node at indices 2i+1 and 2i+2, respectively, where i is the index of the node. However, this implementation is less efficient than a pointer-based implementation because it requires extra space for unused nodes and requires more complex indexing computations.

#### What is the maximum height of an AVL tree with n nodes?

The maximum height of an AVL tree with n nodes is O(log n), which is achieved when the AVL tree is perfectly balanced.

#### How is the height of an AVL tree related to its number of nodes?

The height of an AVL tree is logarithmic in its number of nodes. Specifically, the height of an AVL tree with n nodes is O(log n).

#### Can an AVL tree be used to store data that is not comparable?

No, an AVL tree can only be used to store data that is comparable, i.e., data that can be ordered according to a well-defined relation such as less than or greater than.

#### What is the AVL tree rebalancing algorithm?

The AVL tree rebalancing algorithm is a set of operations that restore the AVL property after a node insertion or deletion has violated it. The algorithm involves performing rotations on the affected nodes and their ancestors until the AVL property is restored.

#### What is the advantage of using an AVL tree over a binary search tree?

The main advantage of using an AVL tree over a binary search tree is that AVL trees guarantee logarithmic height, which ensures efficient search, insertion, and deletion operations in the worst case. Binary search trees, on the other hand, can have linear height in the worst case, which leads to inefficient operations.

#### How does the AVL tree insertion algorithm differ from the binary search tree insertion algorithm?

The AVL tree insertion algorithm differs from the binary search tree insertion algorithm in that it performs additional operations to maintain the AVL property after inserting a new node. Specifically, it rebalances the tree by performing rotations on the affected nodes and their ancestors until the AVL property is restored.

#### Can an AVL tree be unbalanced?

No, an AVL tree cannot be unbalanced because it is designed to maintain the AVL property, which ensures that the heights of the left and right subtrees of any node differ by at most one.