1. What is a Tree
Figure: fig1.
• A data structure similar to linked list.
• Each node points many other.
• Hierarchical nature in graphical form.
2. Features and types
• Root - node with no parent.
• Edge - link from parent to child.
• Leaf node - node with no child.
• Siblings - children of same parent.
• node p is an ancestor of node q if on path from root to q p
appears.
Figure: fig2.
3. conti.
• Level-set of nodes at a given depth (root-0)
Figure: fig3.
• Depth - length of path from rooot to node.
• Height - length of path from node to the deepest node.
• Height of tree - Maximum of all heights.
• Depth of tree - Maximum of all depths.
4. conti.
Conti.
• Size - Number of descendants including itself.
• Skew Trees - Every node in tree has one child.
a. Left Skew tree - only left child.
b. Right Skew tree - only right child.
Figure: fig4.
5. About Binary Trees
• Binary Tree
Binary tree -ro,one or two children. Essentially root and two
disjoint binary trees .
Figure: fig5.
6. Types Of Binary Trees
• Types of Binary Trees
1. Strict Binary Tree - Each node exactly two or no children.
Figure: fig6.
1. Full Binary Tree - Each node 2 children, all leaf nodes same
level.
Figure: fig6.2.
1. Complete Binary Tree - every level, except possibly the last, is
completely filled, and all nodes are as far left as possible..
8. Infering from table
• Conti
We infer:
• No. of nodes = 2h+1 − 1.
• No. of nodes in complete tree are greater than 2h (min.) and
2h+1 − 1(max.)
• No. of leaf nodes in a full binary tree = 2h
• No. of NULL pointers in a complete tree = n + 1(n - total
nodes)
9. Structure of Binary Tree
• Structure of Binary Tree
Figure: fig8.
struct BinaryTreeNode
{
int data;
struct BinaryTreeNode* left;
struct BinaryTreeNode* right;
};
10. Operations on Binary Trees
• Operations on Binary Trees
Basic Operations
* Inserting an element into tree.
* Deleting an element from tree.
* Searching for an element.
* traversing the tree.
Auxiliary Operations
* Finding size of the tree.
* Finding height of tree.
* Finding the level which has maximum sum.
* Finding least common ancestor(LCA).
11. applications
• Applicatons of Binary Tree
• Expression trees - used in compilers.
• Huffman coding trees - used in data compression algorithms.
• Binary Search Trees - support search,insertion and deletion in
O(logn) (average).
• Priority Queues - support search and deletion in logarithmic
time(worst case).
