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Tree

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.

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TREE PRESENTED BY SIMRAN KAUR
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
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)
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
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
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
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
PREORDER TREE TRAVERSAL  Process the root R  Traverse the left subtree of R in preorder  Traverse the right subtree of R in preorder
INORDER TREE TRAVERSAL  Traverse the left subtree of R in inorder  Process the root R  Traverse the right subtree of R in inorder
POSTORDER TREE TRAVERSAL  Traverse the left subtree of R in postorder  Traverse the right subtree of R in postorder  Process the root R
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
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
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
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)
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
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
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
FEASIBILITY  A feasible set is promising, if it can be extended to produce not merely as solution, but an optimal solution to the problem
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
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
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
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
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)
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
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
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
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
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