CSE3358 Problem Set 8
03/18/05
Due 03/25/05
Problem 1: Practice Insertions (10 points)
Show the results of insering the following keys
F, S, Q, K, C, L, H, T, V, W, R, N, P, A, B, X, Y, D, Z, E
in the order shown into an originally empty
(a) (5 points) Red-Black Tree
(b) (5 points) B-tree with t= 2
Problem 2: AVL tree (30 points)
In 1962, Adelson-Velskii and Landis invented the AVL tree. An AVL tree is a binary search tree that
satisfies the following additional property:
AVL property: For any node x,|h[lef t[x]] −h[right[x]]| ≤ 1 where h[x] denotes the height of x, and
we define h[NI L] = −1.
In other words, the heights of the left and right subtrees (children) of a node differ by at most one.
(a) (5 points) Let S(h) be the minimum number of nodes in an AVL tree of height h. Show that
S(0) = 1 and S(1) = 2 and that for h≥2, S(h) = S(h−1) + S(h−2) + 1.
(b) (5 points) Using induction, show that S(h)≥φh, where φ=1+√5
2.
(c) (5 points) Conclude that the height hof an AVL tree on nnodes satisfies h= Θ(log n).
(d) (15 points) Given that every node xstores its height h[x], an AVL tree can be maintained after an
insertion operation. Here’s how: After inserting a new node xas a leaf, we set h[x] = 0 and we go up
the tree updating the height of every node on the path from xto the root by incrementing its height
by 1. When we encounter the first node yfor which |h[left[y]] −h[right[y]]|= 2 (if any), we fix the
AVL property for yby performing rotations. Depending on the position of xrelative to y, different
rotation(s) must be performed. We have two cases to consider (the other cases are symmetric)
•case 1: xis in the left subtree of lef t[y]
•case 2: xis in the right subtree of lef t[y]
The above two cases suggest that, since node yviolates the AVL property after insertion, the left
subtree of yis higher than the right subtree of y(the insertion was done in the left subtree of y),
and the height difference is 2 (more than 1). For each of the two above cases, describe the rotation(s)
required to re-establish the AVL property for node y. Argue that after re-establishing the AVL property
for node y, no more height updates or fixes will be needed, i.e. the resulting tree is AVL.
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