Data Structures and Algorithms
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1 Data Structures and Algorithms Trees Sidra Malik
2 Tree? In computer science, a tree is an abstract model of a hierarchical structure A tree is a finite set of one or more nodes such that: There is a specially designated node called the root. The remaining nodes are partitioned into n>=0 disjoint sets T1,..., Tn, where each of these sets is a tree. We call T1,..., Tnthe subtreesof the root. 2
3 Tree Terminology Trees are a hierarchical data structure of nodes Nodes are linked by edges 4 nodes 2 5 edge 1 3 3
4 Applications Organization charts File systems Programming environments A class hierarchy in programming languages that support single inheritance (e.g. Java) Document Object Model (for HTML and XML Artificial Intelligence (Decision making etc) 4
5 Hierarchical Classification A tree is hierarchical classification. Data items appear at various levels within the organization E.g., directory structure: 5
6 Decision Trees 6
7 Decision Tree based upon Expert System 7
8 Tree Terminology Parent / Child Root: node without parent A parent node references one or more nodes (children nodes) that are lower in the tree hierarchy If node u is the parent of node v, then v is the child of u Except for the root (no parent), every node has exactly one parent (by definition) A tree has only one root node v u root 8
9 Tree Terminology Siblings Two nodes that are children of the same parent. 9
10 Tree Terminology Internal node A node is internal if it has one or more children 10
11 Tree Terminology Leaf /External node A node is a leaf if it has no children 11
12 Tree Terminology Ancestor / Descendent An ancestor of a node is either the node s parent or any ancestor of the node s parent (this is recursive) The root is an ancestor of each other node Descendant of a node: child, grandchild, grandgrandchild v u u v 12
13 Tree Terminology Subtree A tree may be divided into subtrees. A subtree is a tree that has the child of a node as its root. Hence, a subtree is composed of a node and all of that node s descendants. The first node in a subtree is known as the root of the subtree and is used to name the subtree. Subtrees themselves can be further divided into other subtrees. 13
14 Tree Terminology Subtree The subtree of T rooted at node v is the tree consisting of all the descendents of v in T (including v) tree consisting of a node and its descendants v Root of subtree v 14
15 Tree Terminology Depth of a node The depth of a node v in T is the number of ancestors of v, excluding v itself. More formally: If v is the root, the depth of v is 0 v v depth of v = 1 depth of v = 3 15
16 Tree Terminology Depth of a Tree The depth of a tree is the maximum depth of any of its leaves maximum levels of a tree tree depth = 3 tree depth = 2 tree depth = 0 16
17 Tree Terminology Height of a Tree The Height of a tree is the maximum number of nodes possible in a path starting from root to leaf H= max levels + 1 height= 1 tree height= 3 Tree height= 4 17
18 Terminology Height of the tree? Depth of node B? 18
19 Terminology Parents: Children: Siblings: Leaves: Internal Nodes: 19
20 Terminology Parents: A, B, F Children: B, E, F, C, D, G, H, I Siblings: {B, E, F}, {C, D}, {G, H, I} Leaves: C, D, E, G, H, I Internal Nodes: A, B, F 20
21 Terminology Two nodes are adjacentif a branch connects them. A pathis a sequence of nodes in which each node is adjacent to the next one. Every node in the tree can be reached by following a unique path starting from the root. The lengthof this path is the number of edges on the path. There is a path of length 0 from every node to itself. 21
22 Terminology The path from the root, A, to the leaf, I, is denoted as AFI and has a length of 2. ABD is the path from the root, A, to the leaf, D, and also has a length of 2. 22
23 Types of Trees General tree a node can have any number of children Binary tree a node can have at most two children 23
24 Binary Tree
25 Binary Trees The simplest form of treeis a Binary Tree A Binary Tree consists of Twhich is empty A node(called the rootnode) and Disjoint Left and right subtrees, T1and T2 Both the subtrees are themselves binary trees Note: this is a recursive definition (A node can t have more than 2 children) Binary tree General tree 25
26 Binary Trees Difference b/w Trees and Binary Trees A Tree can never be empty but a binary tree may be empty. Binary Tree can t have more than 2 children whereas in case of a tree a node may have any number of children Binary tree General tree 26
27 Binary Tree A binary tree is either empty or has the following form: root T L T R Where Tleft and Tright are binary trees. 27
28 Binary Trees Full binary tree: is said to be full if it contains maximum possible number of nodes in all levels Complete binary tree: is said to be complete if it contains maximum possible number of nodes except possibly the last level 28
29 Binary Trees 29
30 Binary Trees Skewed binary tree: Contains only left or right children. Similar: Two trees with same structure and different data. Copy or identical: Same structure and same data. 30
31 Properties of Binary Trees Max number of nodes on level l is 2 l, l >= 0 If h= height of a binary tree, max number of nodes possible are =2 h -1 Minimum number of nodes possible in binary Tree are h For any non empty binary Tree, ifn isnumber of nodes and e areedges then n=e+1 A binary tree with height hand2 h -1 nodes is called a full binary tree Binary tree 31
32 Binary Tree Structure The representation of a binary tree structure is relatively straightforward. We need a variable to store the data at the node and 2 pointers to the left and right subtrees. struct Node { int data Node *left Node *right } 32
33 Binary Tree Structure 33
34 Binary Tree Operations Insertion Into empty or existing Tree Deletion From a non empty Tree Traversal To visit all nodes in Binary Tree Merge To merge two binary trees into a larger one 34
35 Visiting and Traversing a Node Many applications require that all of the nodes of a tree be visited. Visiting a node may mean printing contents, retrieving information, making a calculation, etc. Traverse:To visit all the nodes in a tree in a systematic fashion. A traversal can pass through a node without visiting it at that moment. 35
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