Question

Red black trees Perform insertions of the following keys, 4, 7, 12, 15, 3, 5, 14, 18, 16, 17 (left to right) into a redblack tree, then, perform deletions of keys 3, 12, 17, under the properties as provided below. • Root propoerty: the root is black. • External propoerty: every leaf is black. • Internal propoerty: the children of a red node are black. • Depth propoerty: all the leaves have the same black depth. Note that insertions have to deal with possible double-red problem and deletions have to deal with possible double-black problem as described in the following. Show the resulting snapshot of the tree after each operation. Consider a double red with child z and parent v, and let w be the sibling of v and there are two possible cases as follows. case 1: w is black, then restructure those 4 nodes to fix the violation. case 2: w is red, then perform a recoloring. Consider cases how to remedy a double black node with sibling y while deletion is performed. case 1: y is black and has a red child. Then, perform a restructuring and then it is done. case 2: y is black and its children are both black. Then, perform a recoloring, which may propagate up the double black violation. case 3: y is red. Then, perform an adjustment after which either case 1 or case 2 applies.

Answer 1

here the answer with solution
12 15 5 17 13 10 14 11 7 4

1. (4B) 2. (4B (7R) 3. (4B (7R (12R))) - whoops, need to rotate left ((4R) 7B (12R)) 4. ((4R) 7B (12R (15R))) - whoops, need to recolor ((4B) 7B (12B (15R))) 5. (((3R) 4B) 7B (12B (15R))) 6. (((3R) 4B (5R) 7B (12B (15R)) 7. (((3R) 4B (5R) 7B (12B ((14R) 15R)) - whoops, need RL rotation ((3R) 4B (5R)) 7B ((12R) 14B (15R))) 8. (((3R) 4B (5R)) 7B ((12R) 14B (15R (18R)))) - whoops, recolor (((3R) 4B (5R)) 7B ((12B) 14R (15B (18R)))) 9. (((3R) 4B (5R)) 7B ((12B) 14R (15B ((16R) 18R)) - whoops, need RL rotation (((3R) 4B (5R)) 7B ((12B) 14R ((15R) 16B (18R)) 10. (((3R) 4B (5R)) 7B ((12B) 14R ((15R) 16B (17R) 18R))) - whoops, need to recolor (((3R) 4B (5R)) 7B ((12B) 14R (15B) 16R ((17R) 18B))) -- whoa, the recolor caused a problem higher up in the tree! --we need to do a left rotation (((3R) 4B (5R)) 7R (12B)) 14B (15B) 16R ((17R) 18B))