Lecture 06: Tree Structures
Topics:
• Trees in general
• Binary Search Trees
• Application: Huffman Coding
• Other types of Trees
Part I. Trees
• Linked Lists
– add [in-order]: O(n)
– remove: O(n)
– find: O(n)
• Trees:
– add [in-order]: O(log n) to O(n)
– remove: O(log n) to O(n)
– find: O(log n) to O(n)
• Try it for these values of n: 4, 8, 1000, 1000000
Basic Idea and terminology
• K-ary tree. This is a 3-ary tree
• depth. The number of nodes above and including a node. depth(bottomleft)=4, depth(bottom-right)=3
• Child: a node which is immediately reachable from a node.
• Descendent: a node which can be reached from a node (regardless of
depth)
• Sibling. Nodes at the same level. There are 3 siblings all descendent from
the root.
• Leaf Node: A node without children
• Interior Node: A node with at least one child.
More terminology
• Each node is a subtree.
• Full tree: all nodes have 0 or k children.
• Perfect Tree: a full tree in which all leaves are at the
same depth
– Very desirable.
– Minimizes depth (makes all operations faster)
• Complete tree: the depth of all ancestors differs by at
most one.
• Degenerate Tree: The opposite of a perfect tree
– Really a…
– …linked list (that's why the worst case O-speed is O(n))
Part II: Binary Search Trees
• A 2-ary tree
– 2 children. Usually called left and right.
• Often represents sorted data
– All children and ancestors in the left branch are
less than (or equal) to this node.
– All children and ancestors in the right branch are
greater than (or equal) to this node.
Operations
• [Insertions]
– [8, 4, 12, 5, 1, 11, 99]
– [1, 4, 5, 11, 8, 12, 99]
• [Finding]
– How many operations to find the 99 in both trees?
• [Removing]
– With no children
– With 1 child
– With 2 children
• Finding the minimum value in the right sub-tree.
• Finding the maximum value in the left sub-tree.
• Traversal: Depth-First Search
– Pre-order, In-order, Post-order
• [Try to code at least some of it]
Part III: Huffman Coding
• Reference:
http://en.wikipedia.org/wiki/Huffman_coding
• ASCII encoding of characters
• Frequency distribution
• Huffman Algorithm
– Developed by David Huffman for PhD thesis at
MIT (1952)
Part IV: Other types of Trees
• B+ Trees
– Each node is (small) array of values
– Siblings point to the next sibling
– Used for indexing in Databases (Access, MySQL,
Oracle, etc)
– Used in many filesystems (NTFS for one)
• Red-Black Tree
– Self-balancing tree (much more complex insertions
and removals) => guaranteed an O(log n) access time.
– Applications:
• computational geometry (Maya, Blender)
• some process schedulers in Linux
• Used in some implementation of STL maps (like python
dictionaries)
• …