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(bottom-

left)=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)

• …