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Binary Search Trees Binary Search Trees I think that I shall never see a poem as lovelyasatree Po ems are wrote byfools likemebut only Gd can makea tree Joyce Kilmer Binary search trees provide a data structure which ef ciently supp orts all six dictionary op erations A binary tree is a ro oted tree where each no de contains at most two children Each child can be identied as either a left or right child parent left right A binary tree can be implemented where each no de has left and right p ointer elds an optional parent p ointer and a data eld Binary Search Trees A binary search tree lab els each no de in a binary tree with a single key such that foranynode x and no des in the left subtree of x have keys x and all no des in the right subtree of x have keys x 2 3 7 8 68 7 5 5 6 2 3 Left A binary search tree Right A heap but not a binary search tree The search tree lab eling enables us to nd where any key is Start at the ro ot if thatisnottheonewewant search either left or right dep ending up on whether what wewant is or then ro ot Searching in a Binary Tree Dictionary search op erations are easy in binary trees TREESEARCHx k if x NILork keyx then return x if kkeyx then return TREESEARCHleftxk else return TREESEARCHrightxk The algorithm works b ecause b oth the left and right subtrees of a binary search tree are binary search trees recursive structure recursive algorithm This takes time prop ortional to the height of the tree O h Maximum and Minimum Where are the maximum and minimum elements in a binary tree TREEMAXIMUMX while rightx NIL do x rightx return x TREEMINIMUMx while lef tx NIL do x leftx return x Both take timeprop ortional to the height of the tree O h Where is the predecessor Where is the predecessorofano de in a tree assuming all keys are distinct X PREDECESSOR(X) SUCCESSOR(X) If X has twochildren its predecessoristhemaximum value in its left subtree and its successortheminimum value in its right subtree predecessor(x) X If it do es not have a left child a no des predecessoris its rst left ancestor The pro of of correctness comes from lo oking at the inorder traversal of the tree H A F B G D CE TreeSuccessorx if rightx NIL then return TreeMinimumrightx y px while y NIL and x righty do x y y py return y Tree predecessorsuccessor b oth run in time prop or tional to the height of the tree InorderTreewalkx if xNIL then InorderTreeWalklef tx print keyx InorderTreewalkrightx ABCDEFGH Tree Insertion Do a binary search to nd where it should be then replace the termination NIL p ointer with the new item LEAF INSERTION FIGURE TreeinsertT z y NIL x r ootT while x NIL do y x if keyz keyx then x lef tx else x rightx pz y if y NIL then rootT z else if keyz keyy then lef ty z else righty z y is maintained as the parent of x since x eventually b ecomes NIL The nal test establishes whether the NIL was a left or right turn from y Insertion takes timeprop ortional to the height of the tree O h Tree Deletion Deletion is somewhat more tricky than insertion be cause the no de to die maynot b e a leaf and thus eect other no des Case a where the no de is a leaf is simple just NIL out the parents child p ointer Case b where a no de has one chld the do omed no de can just b e cut out Case c relab el the no de as its successorwhich has at most one child when z has two children and delete the successor This implementation of deletion assumes parent p oint ers to make the co de nicer but if you had to save space they could be disp ensed with by keeping the p ointers on the search path stored in a stack TreeDeleteT z if lef tz NILorrightz NIL then y z else y TreeSuccessorz if lef ty NIL then x lef ty else x righty if x NIL then px py if py NIL then rootT x else if y lef tpy then lef tpy x else rightpy x if yz then keyz keyy If y has other elds copy them to o return y Lines determine which no de y is physically removed Lines identify x as the nonnil decendant if any Lines give x anewparent Lines mo dify the ro ot no de if necessary Lines reattach the subtree if necessary Lines if the removed no de is deleted copy Conclusion deletion takes time prop ortional to the height of the tree Balanced Search Trees All six of our dictionary op erations when implemented with binary search trees take O h where h is the height of the tree The b est height we could hop e to get is lg n if the tree was p erfectly balanced since P blg nc i n i But if we get unlucky with our order of insertion or deletion we could get linearheight inserta insertb insertc insertd A B C D In fact random search trees on average have lg N height but weare worried ab out worst case height We cant easily use randomization Why Perfectly Balanced Trees Perfectly balanced trees require a lot of work to main tain 9 513 3 7 11 15 2 4 6 8 10 12 14 16 1 If weinsertthekey wemust move every single no de in the tree to rebalance it taking ntime Therefore when we talk ab out balanced trees we mean trees whose height is O lg n so all dictionary op erations insert delete search minmax succes sorpredecessor take O lg n time RedBlack trees are binary search trees where each no de is assigned a color where the coloring scheme helps us maintain the height as lg n.
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