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Chapter 10 Search Trees

Chapter 10 Search Trees. 10.1 Binary Search Trees. 10.1 Binary Search Trees. 6. <. 2. 9. >. =. 8. 1. 4. Dr Zeinab Eid. A binary search tree is a proper binary tree storing keys (or key-value entries) at its internal nodes and satisfying the following property:

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Chapter 10 Search Trees

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  1. Chapter 10Search Trees 10.1 Binary Search Trees Search Trees

  2. 10.1 Binary Search Trees 6 < 2 9 > = 8 1 4 Dr Zeinab Eid Search Trees

  3. A binary search tree is a proper binary tree storing keys (or key-value entries) at its internal nodes and satisfying the following property: Let u, v, and w be three nodes such that u is in the left subtree of v and w is in the right subtree of v. We have key(u)key(v) key(w) External nodes do not store items An inorder traversal of a binary search trees visits the keys in increasing order 6 2 9 1 4 8 10.1 Binary Search Trees Search Trees

  4. 10.1.1 Search AlgorithmTreeSearch(k, v) ifT.isExternal (v)then returnv if k<key(v) returnTreeSearch(k, T.left(v)) else if k=key(v) returnv else{ k>key(v) } returnTreeSearch(k, T.right(v)) • To search for a key k, we trace a downward path starting at the root • The next node visited depends on the outcome of the comparison of k with the key of the current node • If we reach a leaf, the key is not found and we return null • Example: find(4): • Call TreeSearch(4,root) 6 < 2 9 > = 8 1 4 Search Trees

  5. 10.1.2 Update Operations • Insertion • Removal Dr Zeinab Eid Search Trees

  6. Insertion • Let us assume a proper binary tree T supports the following update operation: insertAtExternal(v,e):Insert the element e at the external node v, and expand v to be internal, having new (empty) external node children; an error occurs if v is an internal node. Dr Zeinab Eid Search Trees

  7. Insertion 6 < • To perform operation insert(k, e), we search for key k (using TreeSearch) • Assume k is not already in the tree, and let w be the leaf reached by the search • We insert k at node w and expand w into an internal node • Example: insert 5 2 9 > 1 4 8 > w 6 2 9 1 4 8 w 5 Dr Zeinab Eid Search Trees

  8. Removal • We assume that a proper binary tree T supports the following additional update operation: removeExternal(v): Remove an external node v and its parent, replacing v’s parent with v’s sibling; an error occurs if v is not external. Dr Zeinab Eid Search Trees

  9. Removal • To perform operation remove(k), we search for key k: • Assume key k is in the tree, and let let v be the node storing k • If node v has a leaf child w, we remove v and w from the tree with operation removeExternal(w), which removes w and its parent • Example: remove 4 6 < 2 9 > v 1 4 8 w 5 6 2 9 1 5 8 Dr Zeinab Eid Search Trees

  10. Removal (cont.) 1 • We consider the case where the key k to be removed is stored at a node v whose children are both internal • we find the internal node w that follows v in an inorder traversal • we copy key(w) into node v • we remove node w and its left child z (which must be a leaf) by means of operation removeExternal(z) • Example: remove 3 v 3 2 8 6 9 w 5 z 1 v 5 2 8 6 9 Dr Zeinab Eid Search Trees

  11. Example: Delete65: - FindIn-order Successor of 65, which is 76. - Copy the key of this successor node into node to be deleted. - Remove this successor node, which must have external node as its left child. Dr Zeinab Eid Search Trees

  12. Performance Analysis of Binary-Search Tree • All Search, insert and remove algorithms traverse a path from the root to a node in the BST. • In all three algorithms, we spend O(1) time at each node visited. • In the worst case, the number of nodes visited is proportional to the height h of T. • Thus, in a BST of n-nodes, all search, insert and remove methods run in O(h) time. Dr Zeinab Eid Search Trees

  13. Performance Analysis of Binary-Search Tree • A BST, T, is an efficient implementation of a dictionary of n-entries only if the height of T is small. • Best Case: T has height h=log(n+1), leading to a logarithmic-time performance of all dictionary operations. • Worst Case: T has height n, leading to a linear-time performance, as in case of a set of ordered keys. Dr Zeinab Eid Search Trees

  14. Performance • Consider a dictionary with n items implemented by means of a binary search tree of height h • the space used is O(n) • methods search, insert and remove take O(h) time • The height h is O(n)in the worst case and O(log n) in the best case Dr Zeinab Eid Search Trees

  15. Notice That: • The running time of search and update operations in a BST varies dramatically, depending on the tree’s height. • On the average, a BST with n-keys, generated by a random series of insertions and removals, of n keys has an expected height of O(log n) . • Nevertheless, we must not use standard BST in applications, where updates are not random. Dr Zeinab Eid Search Trees

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