ITEC2620 Introduction to Data Structures

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Gervais Farmer
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1 T2620 ntroduction to ata Structures Lecture 4a inary Trees Review of Linked Lists Linked-Lists dynamic length arbitrary memory locations access by following links an only traverse link in forward direction ow to go both ways? Two links/pointers oubly Linked Lists public class oublelink { public int key; public oublelink left; public oublelink right; } left key right oubly Linked Lists list ach link has two pointers an traverse linked list in forward and backward directions 1

2 oubly Linked Lists oubly Linked Lists V list list prev toelete toelete Previous delete needed to pointers toelete.left.right = toelete.right; toelete.right.left = toelete.left; oubly Linked Lists V list inary Tree efinitions binary tree is a structure that is either empty or which consists of one node connected to two disjoint (binary) subtrees disjoint no common nodes toelete 2

3 inary Tree efinitions inary Tree efinitions ach node of a binary tree has a value, a pointer to a left child node, and a pointer to a right child node (pointers may be NULL) node is the parent of its child nodes inary Tree efinitions V inary Tree efinitions V Node is the root Nodes,,, are leaf nodes They have no children 3

4 inary Tree efinitions V inary Tree efinitions V Nodes,,,, are internal nodes They have children Nodes,,, form the left subtree of node inary Tree efinitions V inary Tree efinitions X,,, is the path (of length 3) from node to node Paths connect ancestors to descendants 4

5 inary Tree efinitions X inary Tree efinitions X Nodes, are siblings Nodes, are have depth 1, they are at level 1 in the tree inary Tree efinitions X inary Tree efinitions X This tree has a height of 4 Node has depth 0 minimum-level binary tree has all levels full except (maybe) the last level,,,,,..., 5

6 inary Tree efinitions XV inary Tree efinitions XV full binary tree has all levels full including the last level,, complete binary tree is a mimumlevel binary tree with nodes filled in from the left on the last level,,,,..., Properties Level i of a full binary tree has 2 i nodes full tree of height h has 2 h-1 leaf nodes 2 h-1 1 internal nodes 2 h 1 total nodes ~50% of nodes are leaves in a full tree the height of a full tree is O(logn) mplementation public class inarynode { public inarynode left; public char key; public inarynode right; } 6

7 mplementation mplementation inary Search Trees ST property: or each node (with a key value of K) in the binary tree, ll nodes in the left sub-tree will have key values less than K, ll nodes in the right sub-tree will have key values greater than K inary Search Trees

8 Searching STs f node has same key value Return it f node has larger key value Search the left sub-tree f node has smaller key value Search the right sub-tree Searching STs f ST is balanced, we get binary search full binary search tree has ideal balancing 50% (remaining) values on each side of each node ode for Searching a ST public static inarynode find (inarynode root, int searchkey) { inarynode current = root; } while (current!= null && current.key!= searchkey) { if (current.key > searchkey) current = current.left; else current = current.right; } return current; nalysis for Searching a ST What is the complexity of the find() method? What is the first question we ask? s there a best, worst, and average case? 8

9 nalysis for Searching a ST nalysis for Searching a ST est case Root node O(1) Worst case O(n) Vine, end node nalysis for Searching a ST V verage case epends on the shape of the tree! On average (random/reasonably balanced trees) O(logn) nsertions o find on ST When null, insert new node Newly inserted nodes are always leaf nodes No changes made to existing tree structure 9

10 nsertions nsertions nsert 2 Put it where you would look for it! nsertions V enefits of STs alanced STs have O(logn) worst and average case find like binary search on an array STs have O(1) update like linked lists

11 Readings and ssignments Suggested Readings from Shaffer (third edition) 4.1.5, 5.1, 5.3.1,

ITEC2620 Introduction to Data Structures

ITEC2620 Introduction to Data Structures 9//201 T2620 ntroduction to ata Structures Lecture b nserting and eleting into Linked ata Structures nserting into Linked Lists nsert an element into a linked sorted from smallest to largest.g..insert(newvalue)