12 - Binary Search Trees Data Structures and Algorithms...

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Binary Search Trees Data Structures and Algorithms Andrei Bulatov
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Algorithms – Binary Search Trees 12-2 Binary Search Tree A binary search tree is a binary rooted tree, in which keys satisfy the Binary Search Tree property: Let x be a node in a binary search tree. If y is a node in the left subtree of x, then key[y] key[x]. If y is a node in the right subtree of x, then key[x] key[y] 5 3 7 2 8 5 5 7 8 6 5
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Algorithms – Binary Search Trees 12-3 Inorder Tree Walk Having a binary search tree one can print its content in sorted order Inorder-Tree-Walk(x) if x Nil then do Inorder-Tree-Walk(left[x]) print key[x] Inorder-Tree-Walk(right[x]) To print the entire tree just call Inorder-Tree-Walk(root[T]) It is called inorder because the root is printed between the subtrees A tree walk can also be preorder and postorder
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Algorithms – Binary Search Trees 12-4 Inorder Tree Walk (cntd) Lemma If x is the root of an n-node subtree, then the call Inorder-Tree-Walk(x) takes Θ (n) time Proof Let T(n) denote the running time If x = Nil then T(0) = c, a constant If n > 0 then T(n) = T(k) + T(n – k – 1 ) + d where k is the number of nodes in the left subtree We prove that T(n) = (c+d) n + c For n = 0 we have (c + d) 0 + c = c = T(0)
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Algorithms – Binary Search Trees 12-5 Inorder Tree Walk (cntd) Proof For n > 0 we have T(n) = T(k) + T(n – k – 1) + d = ((c + d) k + c) + ((c + d) (n – k – 1) + c) + d = (c + d) n + c – (c + d) + c + d = (c + d) n + c QED
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12 - Binary Search Trees Data Structures and Algorithms...

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