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Binary Search Trees

Binary Search Trees. Implementing Balancing Operations AVL Trees Red/Black Trees Reading: 11.5-11.6. Implementing Balancing Operations. Knowing rotations, we must know how to detect that the tree needs to be rebalanced and which way There are 2 ways for tree to become unbalanced

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Binary Search Trees

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  1. Binary Search Trees • Implementing Balancing Operations • AVL Trees • Red/Black Trees • Reading: 11.5-11.6

  2. Implementing Balancing Operations • Knowing rotations, we must know how to detect that the tree needs to be rebalanced and which way • There are 2 ways for tree to become unbalanced • By insertion of a node • By deletion of a node • There are two mechanisms for detecting if a rotation is needed and which rotation to perform: • AVL Trees • Red/Black Trees • It is best for both to have a parent reference in each child node to backtrack up the tree easily

  3. Implementing Balancing Operations • AVL trees (after Adel’son-Vel’ski and Landis) keep a balance factor attribute in each node that equals the height of the right sub-tree minus the height of the left sub-tree • Each time an insertion or deletion occurs: • The balance factors must be updated • The balance of the tree must be checked from the point of change up to and including the root • If a node’s balance factor is > +1 or < -1, the sub-tree with that node as root must be rebalanced

  4. Implementing Balancing Operations • If the balance factor of a node is -2 • If the balance factor of its left child is -1 • Perform a right rotation • If the balance factor of its left child is +1 • Perform a leftright rotation • If the balance factor of a node is +2 • If the balance factor of its right child is +1 • Perform a left rotation • If the balance factor of its right child is -1 • Perform a rightleft rotation

  5. AVL Tree Right Rotation Initial Tree After Add 1 Node is -2 Left Child is -1 7 (-1) 7 (-2) 5 (0) 9 (0) 5 (-1) 9 (0) 3 (0) 6 (0) 3 (-1) 6 (0) 1 (0)

  6. AVL Tree Right Rotation After Right Rotation 5 (0) 3 (-1) 7 (0) 1 (0) 6 (0) 9 (0)

  7. AVL Tree Rightleft Rotation Initial Tree After Remove 3 10 (1) 10 (2) Node is +2 Right Child is -1 5 (-1) 15 (-1) 5 (0) 15 (-1) Remove 3 (0) 13 (-1) 17 (0) 13 (-1) 17 (0) 11 (0) 11 (0)

  8. AVL Tree Rightleft Rotation After Right Rotation After Left Rotation 10 (2) 13 (0) 5 (0) 13 (1) 10 (0) 15 (1) 11 (0) 15 (1) 5 (0) 11 (0) 17 (0) 17 (0)

  9. Red/Black Trees • Red/Black Trees (developed by Bayer and extended by Guibas and Sedgewick) keep a “color” red or black for each node in the tree • The root is black • All children of a red node are black • Every path from the root to a leaf contains the same number of black nodes • The maximum height of a Red/Black tree is roughly 2*log n (not as well controlled as an AVL tree), but the longest path is still O(log n)

  10. Red/Black Tree Insertion • When inserting a new node, it is put in the normal place by its value and its color is always set to red • At the end of insertion, the root is always set to black • While current node’s parent is not the root and is red: • If the color of the parent’s sibling is red, no rotation is needed, but re-coloring is needed from current node • Parent’s color is set to black • Parent’s sibling’s color is set to black • Grandparent’s color is set to red • Current node is set to point to the grandparent for loop

  11. Red/Black Tree Insertion • Insertion – No Rebalancing After no rebalancing - just re-coloring the tree After Insertion of an 8 Initial Tree 7 7 7 4 10 4 10 4 10 8 8 Current Current

  12. Red/Black Tree Insertion • Insertion – No Re-balancing After no rebalancing - just re-coloring the tree Initial Tree After Insertion of a 5 15 15 15 Current 7 20 7 20 7 20 4 10 4 10 4 10 5 5 Current Note that AVL tree logic would have rebalanced!

  13. Red/Black Tree Insertion • (Note that we only continue if parent is red) • If the color of the parent’s sibling is black (or null), the process gets more complex • It is symmetric based on current’s parent’s sibling being the grandparent’s left or right child and current being its parent’s left or right child

  14. Red/Black Tree Insertion • If sibling is left child • If current is also left child, set current to parent and rotate current right around parent • Recolor and rotate left around grandparent • If sibling is right child • If current is also right child, set current to parent and rotate current right around parent • Recolor and rotate right around grandparent

  15. Red/Black Tree Insertion • Insertion with Rebalancing (Example 1) After insertion of a 6 and re-coloring sub-tree After rebalancing (Left rotation around 4) Initial Tree 15 15 15 Current 7 20 7 20 7 20 4 10 4 10 5 10 5 5 4 6 Parent of Current is black so that terminates the loop Current 6

  16. Red/Black Tree Insertion • Insertion with Rebalancing (Example 2) After right rotation and re-coloring sub-tree Initial Tree After insertion of a 5 15 15 15 7 20 7 20 7 20 4 10 4 10 4 10 6 6 5 5 Current 6

  17. Red/Black Tree Insertion • Insertion with Rebalancing (Example 2) After left rotation around 4 Move current to grandparent 15 Current 7 20 5 10 4 6 Parent of Current is black so that terminates the loop

  18. Red/Black Tree Removal • Start loop with current equal to found node • Terminate loop when current is equal to root or color of current is red • If current’s sibling is red, rotate the sibling right around current’s parent • If both of sibling’s children are black, recolor • Else if sibling’s left child is black, rotate right child left around sibling • Rotate sibling right around current’s parent

  19. Red/Black Tree Removal • Removal with Rotations Intermediate Step 1 Recolor for right rotation Initial Tree Sibling is Red 15 Sibling 15 7 20 7 20 4 10 4 10 Current Element to be removed 5 12 5 12 Corresponds to upper right of Figure 11.19

  20. Red/Black Tree Removal • Removal with Rotations Intermediate Step 3 Sibling’s left child is null/Black but both children were not black Intermediate Step 2 After right rotation 7 7 4 15 4 15 5 10 20 5 10 20 Sibling 12 Current Sibling 12 Current Corresponds to lower left of Figure 11.19

  21. Red/Black Tree Removal • Removal with Rotations Intermediate Step 4 After left rotation around sibling Intermediate Step 5 Recolor for right rotation 7 7 4 15 4 15 5 12 20 5 12 20 10 Sibling Current 10 Sibling Current

  22. Red/Black Tree Removal • Removal with Rotations Intermediate Step 6 After right rotation Set current to root to end loop Remove 20 to get to Final Tree Current 7 7 4 12 4 12 5 10 15 5 10 15 20

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