AVL Tree
• Definition:
– binary search tree
– For any node, the heights of left subtree and
right subtree differ at most 1.
– (if differ at most k, called BH(k) tree).
• Theorem (Adel'son-Vel'skii and Landis 1962):
– The height of an AVL tree with N internal
nodes always lies between log (N +1 ) and
1..4404 log(N + 2) - 0.328
AVL Tree Examples
Worst case AVL trees
Figures in these slides are copied from:
http://www.eli.sdsu.edu/courses/fall96/cs660/notes/avl/avl.html#RTFToC2
AVL Tree
• Insertion:
– As normal binary search tree insertion
– Reconstruct tree by rotation if some nodes become unbalanced
• Deletion:
– as normal binary search tree deletion
– The node deleted will be either a leaf or have just one subtree (this is true for all
binary search tree deletion)
– if the deleted node has one subtree, then that subtree contains only one node
– Traverse up the tree from the deleted node checking the balance of each node,
balance the node by rotation if unbalanced
• Theorem:
– An insertion into an AVL tree requires at most one rotation to rebalance a tree. A
deletion may require log(N) rotations to rebalance the tree.
AVL tree insertion examples
Insert into subtree 3
Insert into subtree 2 or 3
Perform single rotation
Perform double rotation
AVL tree deletion examples
Deletion from subtree 1
Perform single rotation
Deletion from subtree 1
Perform double rotation
B-Tree
• Similar to Red-Black tree or others, balanced search
tree
• Different from other balanced search tree, nodes
may have many children. So mainly used for Disk
I/O.
• A node x with n[x] keys
–
–
–
–
has n[x]+1 children,
the keys in x are stored in non-decreasing order,
the keys in children are delimited by the keys
all paths from the root to a leaf are of equal length, i.e. height-balanced
An example of B Tree
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
B-tree of degree t
• Definition:
– Root must have 1 key
– Internal node has at least t-1 keys but at most
2t-1 keys, i.e. has at least t but at most 2t
children.
• Theorem: h≤logt(n+1)/2
• Insertion and deletions:
– More complicate but still log (n)
– Split and merge operation.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
2-3-4 Tree
• A B-tree of degree t=2.
– So every node has 2,3,4 children.
• Recursive definition
–
–
–
–
–
•
Satisfying search tree property
nil is a 2-3-4 tree,
a leaf is a 2-3-4 tree,
an internal node has either 2, 3 or 4 children,
all paths from the root to a leaf are of equal length i.e. a 2-3-4-tree is
height-balanced
Height log(n) in balance search tree of n nodes
– Binary tree or non-binary, such as B-tree
– Whether all n nodes are placed as leaves or also as internal nodes.
• B+-Tree:
– real application data nodes are leaves, internal nodes are keys through
which the search can be traced to real data.
Treap
• Consider as Tree + Heap
-organize the tree as binary search tree by keys
– Assign random chosen priorities to nodes, adjust the tree to obey minheap order property
– i.e. Assume that all keys are distinct, so do priorities, for any node u,
• if v is a left child of u, key[v]<key[u]
• if v is a right child of u, key[v]>key[u]
• If v is a child of u, priority[v]>priority[u]
• As a result, the expected height is log(n),
so are insertion and deletion.
Splay tree
(See the end of amortized analysis)
•
•
•
•
A binary search tree (not balanced)
Height may be larger than log n, even n-1.
However a sequence of n operations takes O(nlog n).
Assumptions: data values are distinct and form a totally
order set
• Operations:
–
–
–
–
–
–
Member(i,S)
Insert(i,S)
Delete(i,S)
Merge(S,S’)
Split(i,S)
All based on
• splay(i,S), reorganize tree so that i to be root if iS, otherwise, the
new root is either max{k S |k<i} or min{k S |k>i}
Splay tree (cont.)
• For examples,
– merge(S,S’)
• Call Splay(, S) and then make S’ the right child
– Delete(i,S), call Splay(i,S), remove I, then
merge(left(i), right(i)).
– Similar for others.
– Constant number of splays called.
Splay tree (cont.)
• Splay operation is based on basic rotate(x)
operation (either left or right).
• Three cases:
– y is the parent of x and x has no grandparent
• rotate(x)
– x is the left (or right) child of y and y is the left
(or right) child of z,
• rotate(y) and then rotate(x)\
– x is the left (or right) child of y and y is the
right (or left) child of z,
• rotate(x) and then rotate(x)
Splay tree (cont.)
• Credit invariant: Node x always has at
least log (x) credits on deposit.
– Where (S)=log (|S|) and (x)=(S(x))
• Lemma:
– Each operation splay(x,S) requires no more
than 3((S)-(x))+1 credits to perform the
operation and maintain the credit invariant.
• Theorem:
– A sequence of m operations involving n inserts
takes time O(mlog(n)).