The document discusses various tree data structures and algorithms. It defines common tree terminology like nodes, roots, leaves. It describes different types of trees like binary trees, binary search trees, and their properties. It explains tree traversal algorithms like preorder, inorder and postorder. It also summarizes minimum spanning tree algorithms like Kruskal's and Prim's, and single source shortest path algorithms like Dijkstra's.

1. TREE
PRESENTED BY
SIMRAN KAUR

2. TREE
 A tree is the data structure that are based on
hierarchical tree structure with set of nodes
 It is an acyclic connected graph with one or
more children nodes and at most one root
node

3. TREE TERMINOLOGY
 NODE: Each data item in a tree
 ROOT: First or top data item in hierarchical
arrangement
 DEGREE OF A NODE: Number of subtrees of
a given node
 DEGREE OF A TREE: Maximum degree of a
node in a tree
 DEPTH OR HEIGHT: Maximum level number
of a node + 1(i.e., level number of farthest leaf
node of a tree + 1)

4. TREE TERMINOLOGY
 NON-TERMINAL NODE: Any node except
root node whose degree is not zero
 TERMINAL NODE OR LEAF NODES: Nodes
having zero
 FOREST: Set of disjoint trees
 PATH: Sequence of consecutive edges from
the source node to the destination node

5. BINARY TREE
 In this kind of tree, the maximum degree of
any node is at most 2
 A binary tree T is defined as a finite set of
elements such that T is empty (called NULL
tree or empty tree) and T contains a
distinguished Node R called the root of T and
the remaining nodes of T form an ordered pair
of disjoint binary trees 𝑇1, and 𝑇2

6. FULL BINARY TREE
 A binary tree in which all leaves are t the
same level or at the same depth and in which
every parent has exactly 0 or 2 children

7. COMPLETE BINARY TREE
A binary tree is one which have the following
properties:
i) Which can have 0, 1 or 2 nodes as a child
node
ii) In which first, we need to fill left node, then
right node in a level
iii) In which, we can start putting data item in
next level only when the previous level is
completely filled

8. PREORDER TREE
TRAVERSAL
 Process the root R
 Traverse the left subtree of R in preorder
 Traverse the right subtree of R in preorder

9. INORDER TREE
TRAVERSAL
 Traverse the left subtree of R in inorder
 Process the root R
 Traverse the right subtree of R in inorder

10. POSTORDER TREE
TRAVERSAL
 Traverse the left subtree of R in postorder
 Traverse the right subtree of R in postorder
 Process the root R

11. BREADTH FIRST TRAVERSAL
(BFT)
 The breadth first traversal of a tree visits the
nodes in the order of depth in the tree
 BFT first visits all the nodes at depth zero
(i.e., root), then all the nodes at depth one
and so on
 At each depth, the nodes are visited from left
to right

12. DEPTH FIRST TRAVERSAL
(DFT)
 Depth first traversal is an algorithm for
traversing or searching a tree, tree structure
or graph
 One starts at the root and explores as far as
possible along each branch before
backtracking

13. BINARY SEARCH TREE
A binary search tree, also called as an ordered
or sorted binary tree, has following properties:
i) The left subtree of a node contains only
nodes with keys less than the node’s key
ii) The right subtree of a node contains only
nodes with keys greater than the node’s key
iii) Both the left subtrees and right subtrees
must also be binary search trees

14. AVERAGE CASE PERFORMANCE
OF BST OPERATIONS
 Internal Path Length (IPL) of a binary tree is
the sum of the depths of its nodes.
 Average internal path length T(n) of binary
trees with n nodes is O(n log n)
 The average complexity to find or insert
operations is T(n) = O(log n)

15. AVL TREES
An AVL (Adelson-Velskii and Land) is a binary
tree with the following additional balance
properties:
i) For any node in the tree, the height of the
left and right subtrees can differ by atmost
ii) The height of an empty subtree is -1

16. AVL TREES
 An AVL is a binary search tree which has the
following properties:
i) The subtrees of every node differ in height
by atmost one
ii) Every subtree is an AVL tree
 Balance factor of a node = Height of left
subtree – Height of right subtree

17. GREEDY ALGORITHMS
 Greedy algorithms are simple and straight-
forward
 They are short sighted in their approach in the
sense they take decisions on the basis of
information at hand without worrying about
the effect of these decisions in the future
 They are used to solve optimization problems

18. FEASIBILITY
 A feasible set is promising, if it can be
extended to produce not merely as solution,
but an optimal solution to the problem

19. SPANNING TREE
 A spanning tree of a graph is any tree that
includes every vertex in the graph
 A spanning tree of a graph G is a subgraph of
G that is a tree and contains all vertices of G
 The number of spanning trees in the complete
graph 𝐾 𝑛 is 𝑛 𝑛−2

20. MINIMUM SPANNING TREE
 A Minimum Spanning Tree (MST) of a
weighted graph G is a spanning tree of G
whose edges sum is minimum weight
 There are two algorithms to find the minimum
spanning tree of an undirected weighted
graph

21. KRUSKAL’S ALGORITHM
 Kruskal’s algorithm is a Greedy algorithm
 In this algorithm, starts with each vertex being
its component. Repeatedly merges two
components into one by choosing the light
edge that connects them, that’s why, this
algorithm is edge based algorithm

22. KRUSKAL’S ALGORITHM
Let G = (V,E) is a connected, undirected,
weighted graph
 Scans the set of edges in increasing order by
weight. The edge is selected such that: i)
acyclicity should be maintained, ii) it should
be minimum weight, iii) when tree T contains
n-1 edges, also must terminate
 Uses a disjoint set of data structure to
determine whether an edge connects vertices
in different components

23. ANALYSIS OF KRUSKAL’S
ALGORITHM
The total time taken by this algorithm to find the
minimum spanning tree is O(E 𝑙𝑜𝑔2 E) (if edges
are already sorted)
But the time complexity, if edges are not sorted
is O(E 𝑙𝑜𝑔2 V)

24. PRIM’S ALGORITHM
 Prim’s algorithm is based on a generic
minimum spanning tree algorithm
 The idea of Prim’s algorithm is to find the
shortest path in a given graph
 The Prim’s algorithm has the property that the
edges in the set A always form a single
connected tree

25. PRIM’S ALGORITHM
We begin with some vertex V in a given graph G
= (V,E), defining the initial set of vertices A.
Then, in each iteration, we choose a minimum
weight edge (u,v), connecting a vertex v in the
set A to the vertex u outside of set A. Then,
vertex u is brought into A. This tree is repeated,
until a spanning tree is formed

26. DYNAMIC PROGRAMMING
ALGORITHMS
 Dynamic programming approach for the
algorithm design solves the problems by
combining the solutions to some problems, as
we do in divide and conquer approach
 It is more powerful
 It is a stage-wise search method suitable for
optimization problems whose solutions may
be viewed as the result of a sequence of
decisions

27. DIJKSTRA’S ALGORITHM
 Dijkstra’s algorithm solves the single source
shortest path problems on a weighted
directed graph
 It is Greedy algorithm
 Dijkstra’s algorithm starts at the source vertex,
it grows a tree T, that spans all vertices
reachable from S

28. THANK YOU!!!