General Tree Concepts Binary Trees

Mark Allen Weiss: Data Structures and Algorithm Analysis in Java General Tree ConceptsBinary Trees Data Structures and Algorithms

Outline • Tree Data Structure • Examples • Definition • Terminology • Binary Tree • Definitions • Examples • Tree Traversal • Binary Search Trees • Definition

Linear Lists and Trees • Linear lists (arrays, linked list)are useful for serially ordered data • (e1,e2,e3,…,en) • Days of week • Months in a year • Students in a class • Trees are useful for hierarchically ordered data • Khalid’s descendants • Corporate structure • Government Subdivisions • Software structure

Examples of trees • directory structure • family trees: • all descendants of a particular person • all ancestors born after year 1800 of a particular person • evolutionary tress (also called phylogenetic trees) • albegraic expressions

Khalid’s Descendants What are other examples of hierarchically ordered data?

Property of a Tree • A treet is a finite nonempty set of nodes • One of these elements is called the root • Each node is connected by an edge to some other node • A tree is a connected graph • There is a path to every node in the tree • There are no cycles in the tree. • It can be proved that a tree has one less edge than the number of nodes. • Usually depicted with the root at the top

Is it a Tree? NO! All the nodes are not connected yes! (but not a binary tree) yes! NO! There is a cycle and an extra edge (5 nodes and 5 edges) yes! (it’s actually the same graph as the blue one) – but usually we draw tree by its “levels”

Subtrees

Tree Terminology • The element at the top of the hierarchy is the root. • Elements next in the hierarchy are the children of the root. • Elements next in the hierarchy are the grandchildren of the root, and so on. • Elements at the lowest level of the hierarchy are the leaves.

Other Definitions • Leaves, Parent, Grandparent, Siblings,Ancestors, Descendents Leaves = {Hassan,Gafar,Sami,Mohmmed} Parent(Musa) = Khalid Grandparent(Sami) = Musa Siblings(Musa) = {Ahmed,Omer} Ancestors(Hassan) = {Amed,Khalid} Descendents(Musa)={Fahad,Sami}

level 0 level 1 level 2 level 3 Levels and Height • Root is at level 0 and its children are at level 1.

Degree of a node • Node degree is the number of children it has

Tree Degree • Tree degree is the maximum of node degrees tree degree = 3

Measuring Trees • The height of a node v is the number of nodes on the longest path from v to a leaf • The height of the tree is the height of the root, which is the number of nodes on the longest path from the root to a leaf • The depth of a node v is the number of nodes on the path from the root to v • This is also referred to as the level of a node • Note that there are slightly different formulations of the height of a tree • Where the height of a tree is said to be the length (the number of edge) on the longest path from node to a leaf

root E A R E Child (of root) S T A Leaves or terminal nodes M P L E Depth of T: 2 Height of T: 1 Trees - Example Level 0 1 2 3

Tree Terminology Example • Ais the root node • Bis the parentof D and E • Cis the siblingof B • Dand E are the children of B • D, E, F, G, I are external nodes, or leaves • A, B, C, Hare internal nodes • The depth, level, orpath length of Eis 2 • The heightof the tree is 3 • The degreeof node Bis 2

C C A F E G G D Tree Terminology Example A path from A to D to G B D subtree rooted at B leaves: C,E,F,G E F G

TreeRepresentation Class TreeNode { Object element; TreeNode firstChild; TreeNode nextSibling; }

Binary Trees • A finite (possibly empty) collection of elements • A nonempty binary tree has a root element and the remaining elements (if any) are partitioned into two binary trees • They are called the left and right subtrees of the binary tree A B C right child of A left subtree of A F D E G right subtree of C H I J

- different when viewed as a binary tree a a - same when viewed as a tree b c c b Difference Between a Tree & a Binary Tree • A binary tree may be empty; a tree cannot be empty. • No node in a binary tree may have a degree more than 2, whereas there is no limit on the degree of a node in a tree. • The subtrees of a binary tree are ordered; those of a tree are not ordered.

Binary Tree for Expressions

L 0 L 1 L 2 L 3 Height of a Complete Binary Tree At each level the number of the nodes is doubled. total number of nodes: 1 + 2 + 22 + 23 = 24 - 1 = 15

Nodes and Levels in a Complete Binary Tree Number of the nodes in a tree with M levels: 1 + 2 + 22 + …. 2M = 2 (M+1) - 1 = 2*2M - 1 Let N be the number of the nodes. N = 2*2M - 1, 2*2M = N + 1 2M = (N+1)/2 M = log( (N+1)/2 ) N nodes : log( (N+1)/2 ) = O(log(N)) levels M levels: 2 (M+1) - 1 = O(2M ) nodes

Binary Tree Properties • The drawing of every binary tree with n elements, n > 0, has exactly n-1 edges. • A binary tree of height h, h >= 0, has at leasth and at most2h-1 elements in it.

maximum number of elements minimum number of elements Binary Tree Properties 3. The height of a binary tree that contains n elements, n >= 0, is at least(log2(n+1)) and at mostn.

Full Binary Tree • A full binary tree of height h has exactly 2h-1 nodes. • Numbering the nodes in a full binary tree • Number the nodes 1 through 2h-1 • Number by levels from top to bottom • Within a level, number from left to right

Node Number Property of Full Binary Tree • Parent of node i is node (i/2), unless i = 1 • Node 1 is the root and has no parent

Node Number Property of Full Binary Tree • Left child of node i is node 2i, unless 2i > n,where n is the total number of nodes. • If 2i > n, node i has no left child.

Node Number Property of Full Binary Tree • Right child of node i is node 2i+1, unless 2i+1 > n,where n is the total number of nodes. • If 2i+1 > n, nodei has no right child.

Complete Binary Tree with N Nodes • Start with a full binary tree that has at least nnodes • Number the nodes as described earlier. • The binary tree defined by the nodes numbered 1 through n is the n-node complete binary tree. • A full binary tree is a special case of a complete binary tree

Example of Complete Binary Tree • Complete binary tree with 10 nodes. • Same node number properties (as in full binary tree) also hold here.

Binary Tree Representation • Array representation • Linked representation

Array Representation of Binary Tree • The binary tree is represented in an array by storing each element at the array position corresponding to the number assigned to it.

Incomplete Binary Trees • Complete binary tree with some missing elements

Example Binary Tree To find a node’s parent use this formula: Parent’s Index = (Child’s Index – 1) / 2 Array index from 0 Ex. To find node’s 2 left and right child Left Child’s Index = 2 * Parent’s Index + 1 Right Child’s Index = 2 * Parent’s Index + 2

Linked Representation of Binary Tree • The most popular way to present a binary tree • Each element is represented by a node that has two link fields (leftChild and rightChild) plus an element field • Each binary tree node is represented as an object whose data type is binaryTreeNode • The space required by an n node binary tree isn * sizeof(binaryTreeNode)

Linked Representation of Binary Tree

Node Class For Linked Binary Tree TreeNode class private class TreeNode { private String data; private TreeNode left; private TreeNode right; public TreeNode(String element) { data = element; left = null; right = null; } }

Common Binary Tree Operations • Determine the height • Determine the number of nodes • Make a copy • Determine if two binary trees are identical • Display the binary tree • Delete a tree • If it is an expression tree, evaluate the expression • If it is an expression tree, obtain the parenthesized form of the expression

Binary Tree Traversal • A traversal of a tree is a systematic way of accessing or "visiting" all the nodes in the tree. • There are three basic traversal schemes: preorder, postorder, inorder • .In any of these traversals we always visit the left subtree before the right

Binary Tree Traversal Methods • Preorder • The root of the subtree is processed first before going into the left then right subtree (root, left, right). • Inorder • After the complete processing of the left subtree the root is processed followed by the processing of the complete right subtree (left, root, right). • Postorder • The root is processed only after the complete processing of the left and right subtree (left, right, root). • Level order • The tree is processed by levels. So first all nodes on level i are processed from left to right before the first node of level i+1 is visited

Preorder Traversal public void preOrderPrint() { preOrderPrint(root); } private void preOrderPrint(TreeNode tree) { if (tree != null) { System.out.print(tree.data + " "); // Visit the root preOrderPrint(tree.left);// Traverse the left subtree preOrderPrint(tree.right); // Traverse the right subtree } }

H D L F N B J A E I C G K M O Preorder Example (visit = print) N-L-R

Preorder of Expression Tree / * + a b - c d + e f Gives prefix form of expression.

Inorder Traversal public void inOrderPrint() { inOrderPrint(root); } public void inOrderPrint(TreeNode t) { if (t != null) { inOrderPrint(t.left); System.out.print(t.data + " "); inOrderPrint(t.right); } }

H D L F N B J A E I C G K M O Inorder Example (visit = print) L-N-R

Inorder by Projection

Inorder of Expression Tree • Gives infix form of expression, which is how we normally write math expressions. What about parentheses?

Postorder Traversal public void ostOrderPrint() { postOrderPrint(root); } public void postOrderPrint(TreeNode t) { if (t != null) { postOrderPrint(t.left); postOrderPrint(t.right); System.out.print(t.data + " "); } }

H D L F N B J A E I C G K M O Postorder Example (visit = print) L-R-N

General Tree Concepts Binary Trees