Search Trees

Binary Search Trees
- Definition (HSM Ch.5.7.1, Fig.5.28)
- Searching a BST by key (HSM Ch.5.7.2, Prog.5.20-21)
- Searching a BST by rank (HSM Ch.5.7.2, Prog.5.22)
- Insertion into a BST (HSM Ch.5.7.3, Fig.5.29, Prog.5.23)
- Deletion from a BST (HSM Ch.5.7.4, Fig.5.29-30, Prog.5.23)
AVL Trees
- The need for balancing BSTs (HSM Ch.10.2, Fig.10.8-10)
- Definitions: Height balanced, Balance factor (HSM p.554, Fig.10.8)
- AVL Tree: For all nodes, balance factor(N) = -1,0, or 1
- Maintaining AVL trees
  - Balance whenever a node has BF +-2
  - Balance wrt nearest parent of inserted node with BF = +-2
  - Example (HSM Fig.10.11)
- Rotations
  - New node in left of left of unbalanced node => do LL
  - New node in right of left of unbalanced node => do LR
  - New node in right of right of unbalanced node => do RR
  - New node in left of right of unbalanced node => do RL
- Analysis (HSM Fig.10.13)
2-3 Trees
- Definitions (HSM p.567)
  - 2-nodes and 3-nodes
  - 2-3 trees
  - Number of nodes between 2^h-1 and 3^h-1
  - Hieght between ceil(log₃(n+1)) and ceil(log₂(n+1))
- Example (HSM Fig.10.14)
- Searching a 2-3 tree = O(lnN)
- Insertion = O((2*)lnN)
  - Into a 2-node (HSM Fig.10.15a)
  - Into a 3-node with a 2-node parent (HSM Fig.10.15b)
  - Into a 3-node with a 3-node parent (HSM Fig.10.16)
- Deletion = (O(lnN))
  - Example (HSM Fig.10.17)
  - If not deleting from a leaf, delete from a leaf (largest in left subtree or smallest in right) to replace.
  - From a leaf 3-node is direct
  - From a leaf 2-node with a 3-node sibling => rotate (HSM Fig.10.18)
  - From a leaf 2-node with a 2-node sibling => combine (HSM Fig.10.19)
B Trees
- Motivation to reduce disk access (HSM p.602)
- Definition (HSM p.603-604)
  - Full m-way search tree
  - Root is at least a 2-node
  - Non-root nodes are at least ceil(m/2)-nodes
- Examples (HSM Fig.10.38-39)
- Insertion = O(h)
  - Search for leaf
  - Insert if space
  - Split on median and insert median in parent
- Deletion = O(h)
  - Delete from leaf
  - If enough left, done
  - If too few left rotate if possible, else combine
- Choice of m
  - Larger m => smaller h => lower complexity
  - Larger m => larger nodes => more data to transfer and search
  - Need to balance based on disk characteristics

Exercises

Deletion of an element from a BST: HSM Ch.5.7, exercise 1.

Lab Tasks

Exam Style Questions

Define a binary search tree.
Give the {iterative,recursive} algorithm for search a binary search tree.
[Picture of some BST, make up your own :-]
Derive, using using step counts or reasonable explanation, the asymptotic time complexity of the {iterative,recursive} algorithm for search a binary search tree.
Describe what extra information has to be stored in each node of a BST to enable the BST to be searched by rank. Outline the algorithm for seaching such an extended BST by rank. What is the asymptotic time complexity of the algorithm?
What is the advantage of an AVL tree over a binary search tree?
Copy the following AVL tree to your answer book, then annotate each node with its balance factor.
[Picture of some AVL tree, make up your own :-]
Given the attached generic rotations for maintaining AVL trees, draw the sequence of AVL trees that result from adding the sequence of values [some new values] to this AVL tree.
[Picture of some AVL tree, make up your own :-]
Give the definition of a 2-3 tree.
What is the asymptotic time complexity of {searching,inserting into, deleting from} a 2-3 tree?
Draw the sequence of 2-3 trees that result from adding the sequence of values [some new values] to this 2-3 tree.
[Picture of some 2-3 tree, make up your own :-]
Draw the sequence of 2-3 trees that result from deleting the sequence of values [some new values] from this 2-3 tree.
[Picture of some 2-3 tree, make up your own :-]
What is the motivation for the use of B-trees?
Give rough (high level) algorithms for insertion and deletion from a B-tree.
Explain the factors that affect the choice of the degree of a B-tree.