Binary trees. Binary trees. Binary trees

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Bethany Matthews
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1 Binary trees March 23, Binary trees A binary tree is a tree in which each internal node has at most two children. In a proper binary tree, each internal node has exactly two children. Children are ordered: either left or right. Applications: Arithmetic expressions Decision trees Search trees March 23, Binary trees Proposition. If T is a binary tree of height h with n nodes of which n e are external and n i are internal, then 1) h + 1 n 2 h+1 1, 2) 1 n e 2 h, 3) h n i 2 h 1 and 4) log 2 (n + 1) 1 h n 1. March 23,

2 The ADT binary tree In addition to all methods of the ADT tree, this ADT has left(p): Return the position of the left child of a position p (or null). right(p): Return the position of the right child of a position p (or null). sibling(p): Return the position of the sibling of a position p (or null). March 23, Interface BinaryTree public interface BinaryTree<E> extends Tree<E> { public Position<E> left(position<e> p) throws IllegalArgumentException; public Position<E> right(position<e> p) throws IllegalArgumentException; public Position<E> sibling(position<e> p) throws IllegalArgumentException; March 23, Linked structures B A D C E March 23,

3 Linked structures B A D C March 23, E Inner class Node protected static class Node<E> implements Position<E> { private E element; private Node<E> parent, left, right; public Node(E e, Node<E> p, Node<E> l, Node<E> r) { element = e; parent = p; left = l; right r;... March 23, Inner class Node protected static class Node<E> implements Position<E> {... public E getelement() { return element; public Node<E> getparent() { return parent; public Node<E> getleft() { return left; public Node<E> getright() { return right; public void setelement(e e) { element = e; public void setparent(node<e> v) { parent = v; public void setleft(node<e> v) { left = v; public void setright(node<e> v) { right = v; March 23,

4 Traversal algorithms A traversal algorithm visits all nodes of a tree in some order. Preorder traversal visits each node before its descendants. Postorder traversal visits each node after its descendants. Both traversals visit descendants of the same depth from left to right. March 23, Preorder traversal Preorder traversal visits each node before its descendants. Algorithm preorder(v) visit(v) for each child w of v preorder (w) March 23, Postorder traversal Postorder traversal visits each node after its descendants. Algorithm postorder(v) for each child w of v postorder (w) visit(v) March 23,

5 Inorder traversal Inorder traversal visits each node after its left descendants but before its right descendants. Algorithm inorder(v) inorder (left(v)) visit(v) inorder (right(v)) March 23, Inorder traversal Inorder traversal visits each node after its left descendants but before its right descendants March 23, Arithmetic expression trees Internal nodes are operators: +,,, /. External nodes are operands: numbers, variables. 2 - a b (2 (a 1) + (3 b)) March 23,

6 Decision trees Internal nodes are yes/no questions. External nodes are decisions. Want a fast meal? Yes No How about coffee? On expense account? Yes No Yes No Bistro Cave Trinity Restaurant Vito s March 23, Printing arithmetic expressions An application of inorder traversal: If there is a left subtree, print ( and recur on its left subtree. Print an operand/operator. If there is a right subtree, recur on its right subtree and print ). March 23, Printing arithmetic expressions An application of inorder traversal: Algorithm printexpression(v) if hasleft (v) print( ( ) printexpression(left(v)) print(v.getelement ()) if hasright (v) printexpression(right(v)) print ( ) ) March 23,

7 Evaluating arithmetic expressions An application of postorder traversal: A recursive method that returns the value of a subtree. When visiting an external node, simply return its value. When visiting an internal node, combine the values of its left and right subtrees. March 23, Evaluating arithmetic expressions An application of postorder traversal: Algorithm evaluateexpression(v) if isexternal(v) return v.getelement() else l evaluateexpression(left(v)) r evaluateexpression(right(v)) v.getelement() return l r March 23,

Trees. Make Money Fast! Bank Robbery. Stock Fraud. Ponzi Scheme Goodrich, Tamassia. Trees 1

Trees. Make Money Fast! Bank Robbery. Stock Fraud. Ponzi Scheme Goodrich, Tamassia. Trees 1 Trees Make Money Fast! Stock Fraud Ponzi Scheme Bank Robbery Trees 1 What is a Tree In computer science, a tree is an abstract model of a hierarchical structure A tree consists of nodes with a parent-child