Ch. 12: Binary Search Trees Ming-Te Chi Algorithms Ch. 12: Binary Search Trees Ming-Te Chi Ch12 Binary Search Trees

12.0 What is a binary tree? Binary tree: Empty, or Composed by 3 disjoint set of nodes: root left subtree (binary tree), could be empty. right subtree (binary tree ), could be empty. Ch12 Binary Search Trees

12.1 What is a binary search tree? Binary tree with the binary-search property. Binary-search 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[y] ≥ key[x]. Ch12 Binary Search Trees

Binary search Tree Ch12 Binary Search Trees

Tree traversal Preorder Inorder Postorder Ch12 Binary Search Trees

Inorder tree walk (traversal) INORDER_TREE_WALK(x) 1 if 2 INORDER_TREE_WALK(x.left) 3 print x.key[x] 4 INORDER_TREE_WALK(x.right) Ch12 Binary Search Trees

Theorem 12.1 If x is the root of an n-node subtree, then the call INORDER-TREE-WALK(x) takes (n) time. Proof: Divide and conquer (recurrence) Substitution method. Ch12 Binary Search Trees

Preorder tree walk Postorder tree walk Ch12 Binary Search Trees

12.2 Querying a binary search tree Ch12 Binary Search Trees

TREE_SEARCH(x, k) TREE_SEARCH(x, k) 1 if or 2 return x 3 if 4 return TREE_SEARCH(x.left, k) 5 else return TREE_SEARCH(x.right,k) Ch12 Binary Search Trees

ITERATIVE_SEARCH(x, k) 1 While and 2 do if 3 then 4 else 5 return x (more efficient) Ch12 Binary Search Trees

Q & A What is the complexity of TREE-SEARCH(x,k)? Ch12 Binary Search Trees

MAXIMUM and MINIMUM TREE_MINIMUM(x) TREE_MAXIMUM(x) 1 while x.left  NIL 2 x = x.left 3 return x TREE_MAXIMUM(x) 1 while x.right  NIL 2 x = x.right Ch12 Binary Search Trees

Q & A What is the complexity of TREE-MINIMUM & TREE-MAXIMUM? Ch12 Binary Search Trees

Successor and predecessor The successor of a node x is the smallest key greater than key[x]. Predecessor: The predecessor of a node x is the (write your own definition.) Ch12 Binary Search Trees

TREE_SUCCESSOR Two cases: Right subtree is empty: (Node 15, X=13) Right subtree is not empty: Minimum element in the right subtree. (node 17, when x=15) Ch12 Binary Search Trees

TREE_SUCCESSOR Right subtree is empty: (the successor is the lowest ancester of x whose left child is also an ancestor of x) Go up the tree from x until we see a node that is the left child of its parent. Remarks: x has no parent (x is the root), no successor. (no element is greater then x) x has parent: x is the left child of its parent, the parent is the successor x is the right child of its parent (x >= parent[x]), moves up one level. (ref. Figure 12.2 on page 257, node 13.) Ch12 Binary Search Trees

TREE_SUCCESSOR TREE_SUCCESSOR 1 if 2 then return TREE_MINIMUM(x.right) 3 4 while and 5 6 7 return y Ch12 Binary Search Trees

Theorem 12.2 The dynamic-set operations, SEARCH, MINIMUM, MAXIMUM, SUCCESSOR, and PREDECESSOR can be made to run in O(h) time on a binary search tree of height h. Ch12 Binary Search Trees

12.3 Insertion and deletion The dynamic-set operations, INSERT and DELETE change the structure of the binary-search-tree. The property of the BST remains. left[x] <= x <= right[x] O(h) operations ? Ch12 Binary Search Trees

Tree-Insert(T, z) 1 y = NIL  trace the parent of x (denoted by y) 2 x = T.root  trace the path 3 while x  NIL  moves down until x is set to NIL 4 y = x 5 if z.key < x.key 6 x = x.left 7 else x = x.right 8 z.p = y  z is inserted at x (p[z] is set to y) 9 if y = NIL 10 T.root[T] = z  tree T was empty 11 else if z.key < y.key 12 y.left = z 13 else y.right = z Ch12 Binary Search Trees

Inserting an item with key 13 into a binary search tree Ch12 Binary Search Trees

Deletion: DELETE(T, z) 3 cases z has no child. (simple) Set (P[z])’s z-child to NIL z has just one child. (left or right) Spice out z z has both a left and a right child: find z’s successor y Spice out z’s successor y (y has no left child) Replace z’s data by y’s data Ch12 Binary Search Trees

DELETE(T,z) - reorganization 3 cases z has no left child. Replace z by its right child z has just one child, which is its left child Replace z by its left child z has both a left and a right child, find z’s successor y 1. If y is z’s right child, replace z by y 2. Otherwise, y lies within z’s right subtree, but is not ’’s right child. Replace y, by its own right child, and then we replace z by y. Ch12 Binary Search Trees

z has no left child Replace z by its right child Ch12 Binary Search Trees

z has just one child, which is its left child Replace z by its left child Ch12 Binary Search Trees

z has two children find z’s successor y If y is z’s right child, replace z by y Ch12 Binary Search Trees

z has two children Otherwise, y lies within z’s right subtree, but is not z’s right child. Replace y, by its own right child, and then we replace z by y. Ch12 Binary Search Trees

TRANSPLANT Transplant(T, z, y) Ch12 Binary Search Trees

Ch12 Binary Search Trees

Theorem 12.3 The dynamic-set operations, INSERT and DELETE can be made to run in O(h) time on a binary search tree of height h. Ch12 Binary Search Trees

Remarks on binary-search-tree All basic operations take time proportional to the height of the tree. A complete binary tree with n nodes, the height is log n, the operations runs in O(log n) worst-case time. Complete tree: all leaves have the same depth and all internal nodes have the same degree. If the tree is linear chain of n nodes, the same operations take O(n) worst-case time. (No better than linked list) Ch12 Binary Search Trees

Remarks on binary-search-tree Expected height of randomly build binary search tree is O(log n). (ref. §12.4) In real life, randomly build binary search tree can not be expected. Variations of binary search tree could achieve this O(log n) height. Red-black tree is one of the examples. Ch12 Binary Search Trees