BST (Binary Search Tree)

If you need to maintain a list of objects that is sorted and unique

& If you need to is able to quickly inserts and retrieve objects to and from the This list--The ideal data structure would be a tree set (or a tree map, if you consider each object a key and associate another Object called a value to it).implementation in Java:treeset<t>, Treemap<k, v>a binary tree is a BST iff, for every node n, in the tree:

• All keys n ' s left subtree is less than the key in N, and
• All keys in n ' s right subtree is greater than the key in N.

Insertion-o (log n)From Root all the "to" leaf, compare and decide which side to go. The new node is always a leaf node. Deletion-o (1)-O (log n)If N has no children, we are only having to remove N from the tree. If N has a single child, we remove N and connect its parent to its child.           If N has both children, we need to:find the smallest node that's larger than N, call it M.          Remove m from the tree, Replace the value of N with M. (Think:m always has no left child) Retrieval-o (log n)For BST (binary search trees), although the Average-caseTimes for theLookup,Insert, andDeleteMethods is all O (log N), where N is the number of nodes in the Tree,the worst-caseTime is O (N).We can guarantee O (log N) time for all three methods by using a Balanced tree -a tree tha t always have height O (log N)--instead of a binary search tree. Balanced tree- AVL tree, 2-4 tree, red-black tree and B trees "fully populated" means that every internal node have exactly the children, and all terminal nodes is at the Same depth.

1 classBST {2     Private classNode {3         intVal;4 Node left;5 Node right;6         7          PublicNode () {}8         9          PublicNode (intval) {Ten              This. val =Val; One         } A          -          Public voidcopy (Node N) { -              This. val =N.val; the              This. left =N.left; -              This. right =N.right; -         } -     } +      -      Public StaticNode Root; +      A     //Insert, O (lg N) at      Public voidInsertintval) { -Root =Insert (Root, Val); -     } -      -     PrivateNode Insert (node node,intval) { -         if(node = =NULL) { innode =NewNode (val); -             returnnode; to         } +          -         if(Node.val > val) node.left =Insert (Node.left, Val); the         if(Node.val < val) Node.right =Insert (Node.right, Val); *          $         returnnode;Panax Notoginseng     } -      the     //Search, O (lg N) +      PublicNode Search (intval) { A         returnSearch (Root, Val); the     } +      -     PrivateNode Search (node node,intval) { $         if(node = =NULL)return NULL; $          -         if(Node.val = = val)returnnode; -         Else if(Node.val > Val)returnSearch (Node.left, Val); the         Else returnSearch (Node.right, Val); -     }Wuyi      the     //Delete, O (1)-O (LG N) -      PublicNode Delete (intval) { WuRoot =Delete (root, Val); -         returnRoot; About     } $      -     PrivateNode Delete (node node,intval) { -         if(node = =NULL)return NULL; -         if(Node.val > val) node.left =Delete (Node.left, Val); A         Else if(Node.val < val) Node.right =Delete (Node.right, Val); +          the         Else { -Node del =NewNode (); $ del.copy (node); the          the             if(Node.left = =NULL) {node.copy (node.right); node.right =NULL;returnnode;} the             if(Node.right = =NULL) {node.copy (node.left); node.left =NULL;returnnode;} the          - node.copy (min (del.right)); inNode.right =deletemin (del.right); theNode.left =Del.left; the         } About         returnnode; the     } the      the     Privatenode min (node node) { +         if(node = =NULL|| Node.left = =NULL)returnnode; -         returnmin (node.left); the     }Bayi      the     //Remove the smallest node and return new root; the     Privatenode Deletemin (node node) { -         if(node = =NULL)return NULL; -         if(Node.left = =NULL) { the             returnNode.right;//node is deleted the         } theNode.left =deletemin (node.left); the         returnnode; -     } the}

BST (Binary Search Tree)