Red Black Trees Definitions and Bottom Up Insertion Red Black Trees Definition A red black tree is a binary search tree in which Every node is colored either Red or Black Each NULL pointer is considered to be a Black node If a node is Red then both of its children are Black Every path from a node to a NULL contains the same number of Black nodes By convention the root is Black Definition The black height of a node X in a red black tree is the number of Black nodes on any path to a NULL not counting X 8 3 2007 UMBC CSMC 341 Red Black Trees 1 2 X A Red Black Tree with NULLs shown Black Height of the tree the root 3 Black Height of node X 2 8 3 2007 UMBC CSMC 341 Red Black Trees 1 3 A Red Black Tree with Black Height 3 8 3 2007 UMBC CSMC 341 Red Black Trees 1 4 X Black Height of the tree Black Height of X 8 3 2007 UMBC CSMC 341 Red Black Trees 1 5 Theorem 1 Any red black tree with root x has n 2bh x 1 nodes where bh x is the black height of node x Proof by induction on height of x 8 3 2007 UMBC CSMC 341 Red Black Trees 1 6 Theorem 2 In a red black tree at least half the nodes on any path from the root to a NULL must be Black Proof If there is a Red node on the path there must be a corresponding Black node Algebraically this theorem means bh x h 2 8 3 2007 UMBC CSMC 341 Red Black Trees 1 7 Theorem 3 In a red black tree no path from any node X to a NULL is more than twice as long as any other path from X to any other NULL Proof By definition every path from a node to any NULL contains the same number of Black nodes By Theorem 2 a least the nodes on any such path are Black Therefore there can no more than twice as many nodes on any path from X to a NULL as on any other path Therefore the length of every path is no more than twice as long as any other path 8 3 2007 UMBC CSMC 341 Red Black Trees 1 8 Theorem 4 A red black tree with n nodes has height h 2 lg n 1 Proof Let h be the height of the red black tree with root x By Theorem 2 bh x h 2 From Theorem 1 n 2bh x 1 Therefore n 2 h 2 1 n 1 2h 2 lg n 1 h 2 2lg n 1 h 8 3 2007 UMBC CSMC 341 Red Black Trees 1 9 Bottom Up Insertion Insert node as usual in BST Color the node Red What Red Black property may be violated Every node is Red or Black NULLs are Black If node is Red both children must be Black Every path from node to descendant NULL must contain the same number of Blacks 8 3 2007 UMBC CSMC 341 Red Black Trees 1 10 Bottom Up Insertion Insert node Color it Red X is pointer to it Cases 0 X is the root color it Black 1 Both parent and uncle are Red color parent and uncle Black color grandparent Red Point X to grandparent and check new situation 2 zig zag Parent is Red but uncle is Black X and its parent are opposite type children color grandparent Red color X Black rotate left right on parent rotate right left on grandparent 3 zig zig Parent is Red but uncle is Black X and its parent are both left right children color parent Black color grandparent Red rotate right left on grandparent 8 3 2007 UMBC CSMC 341 Red Black Trees 1 11 G P X U G X P U Case 1 U is Red Just Recolor and move up 8 3 2007 UMBC CSMC 341 Red Black Trees 1 12 G P U X S X P Case 2 Zig Zag Double Rotate X around P X around G S U Recolor G and X 8 3 2007 G UMBC CSMC 341 Red Black Trees 1 13 G P X U S P X G Case 3 Zig Zig Single Rotate P around G S Recolor P and G 8 3 2007 UMBC CSMC 341 Red Black Trees 1 U 14 Asymptotic Cost of Insertion O lg n to descend to insertion point O 1 to do insertion O lg n to ascend and readjust worst case only for case 1 Total O log n 8 3 2007 UMBC CSMC 341 Red Black Trees 1 15 Top Down Insertion An alternative to this bottom up insertion is top down insertion Top down is iterative It moves down the tree fixing things as it goes What is the objective of top down s fixes 8 3 2007 UMBC CSMC 341 Red Black Trees 1 16 Insert 4 into this R B Tree 11 14 2 1 7 5 Black node 8 3 2007 15 8 Red node UMBC CSMC 341 Red Black Trees 1 17 Insertion Practice Insert the values 2 1 4 5 9 3 6 7 into an initially empty Red Black Tree 8 3 2007 UMBC CSMC 341 Red Black Trees 1 18
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