Question 1: AVL Trees. AVL Trees are yet another self balancing binary search tree (BST) that are sometimes used in the place of red black trees. The key property of an AVL tree is that for all nodes n in the tree, height(n. left) height(n. right) Fh, where Fh is the hih Fibonacci number. (Fo = 1, F1 = 1, F2 = 2, ...) (Hint Use strong induction with two base cases. First establish the property for all AVL trees of heights 0 and 1. Next, assuming it holds for trees of height 30, Fk > 1.5k. (B) Using the above fact and the result from part A, show that h = (log(n)). (C) We will briefly examine inserting a node into an AVL tree through an example. On the left, we have shown an AVL tree and to the right we show the result after a BST insert has happened. AVL Tree Before and After Insertion Devise a sequence of left and right rotations that will restore the AVL tree property. Explain for each rotation what is the root node at which we are rotating and which direction. If you wish, you may insert images showing the trees before/after rotation using markdown (see how we inserted the image. But do not forget to upload the images with the submission). Question 1: AVL Trees. AVL Trees are yet another self balancing binary search tree (BST) that are sometimes used in the place of red black trees. The key property of an AVL tree is that for all nodes n in the tree, height(n. left) height(n. right) Fh, where Fh is the hih Fibonacci number. (Fo = 1, F1 = 1, F2 = 2, ...) (Hint Use strong induction with two base cases. First establish the property for all AVL trees of heights 0 and 1. Next, assuming it holds for trees of height 30, Fk > 1.5k. (B) Using the above fact and the result from part A, show that h = (log(n)). (C) We will briefly examine inserting a node into an AVL tree through an example. On the left, we have shown an AVL tree and to the right we show the result after a BST insert has happened. AVL Tree Before and After Insertion Devise a sequence of left and right rotations that will restore the AVL tree property. Explain for each rotation what is the root node at which we are rotating and which direction. If you wish, you may insert images showing the trees before/after rotation using markdown (see how we inserted the image. But do not forget to upload the images with the submission)