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Binary Search Trees CSE 331 Section 2 James Daly

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Homework 1 Homework 1 is due on today Leave on the front table

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Review: SampleTrees … T1T1 T2T2 TkTk

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Review: Terminology Parent Child Ancestor Descendent Root Leaf Internal A BC D

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Review: Terminology Path Depth Height A BC D

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Review: Binary Tree Every node has at most 2 children Left and right child Variation: n-ary trees have at most n children

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Binary Search Tree For every node Left descendents smaller (l ≤ k) Right descendents bigger (r ≥ k) k

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Traversal Three types Pre-order Visit the parent before either of its children In-order Visit the left children before the parent Visit the right children after Post-order Visit the children before the parent

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Eval Tree *++abc-d3 Pre-order: * +ab +c -d3 In-order: a + b*c + d – 3 Post-order: ab+ cd3- + *

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Search Tree

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All keys in left subtree <= root All keys in right subtree >= root In-order traversal => non-decreasing list Tree-sort O(n log n) build time for balanced trees O(n) time to traverse tree Can define functions to find particular items or the largest or smallest item

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Find(t, x) If (t = null) null Else If (x < t.data) Find(t.left, x) Else If (x > t.data) Find(t.right, x) Else t ?

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FindMin(t) [Recursive] If (t.left != null) t Else FindMin(t.left)

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Insert(t, x) If (t = null) t = new Node(x) Else If (x < t.data) Insert(t.left, x) Else If (x > t.data) Insert(t.right, x) Construct a BST for 5, 3, 9, 2

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Delete: 1 st Case Leaf Node

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Delete: 2 nd Case One Child

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Delete: 3 rd Case Two children Swap with least successor (or greatest predecessor) Then delete from the right (or left) subtree

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Delete: 3 rd Case

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Next Time Balanced binary search trees AVL trees Maybe Red-Black trees

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