Multi-Way search Trees Trees: a. Nodes may contain 1 or 2 items. b. A node with k items has k + 1 children c. All leaves are on same level.

Example A 2-3 tree storing 18 items

Updating Insertion: Find the appropriate leaf. If there is only one item, just add to leaf. Insert(23); Insert(15) If no room, move middle item to parent and split remaining two item among two children. Insert(3);

Insertion Insert(3);

Insert(3); In mid air…

Done…

Tree grows at the root… Insert(45);

New root:

Delete If item is not in a leaf exchange with in- order successor. If leaf has another item, remove item. Examples: Remove(110); (Insert(110); Remove(100); ) If leaf has only one item but sibling has two items: redistribute items. Remove(80);

Remove(80); Step 1: Exchange 80 with in-order successor

Remove(80); Redistribute

Some more removals… Remove(70); Swap(70, 75); Remove(70); “Merge” Empty node with sibling; Join parent with node; Now every node has k+1 children except that one node has 0 items and one child. Sibling can spare an item: redistribute

Delete(70)

New tree: Delete(75) will “shrink” the tree

Details 1. Swap(75, 90) //inorder successor 2. Remove(75) //empty node created 3. Merge with sibling 4. Drop item from parent// (50,90) empty Parent 5. Merge empty node with sibling, drop item from parent (95) 6. Parent empty, merge with sibling drop item. Parent (root) empty, remove root.

“Shorter” 2-3 Tree

Deletion Summary If item k is present but not in a leaf, swap with inorder successor; Delete item k from leaf L. If L has no items: Fix(L); Fix(Node N); //All nodes have k items and k+1 children // A node with 0 items and 1 child is possible, it will have to be fixed.

Deletion (continued) If N is the root, delete it and return its child as the new root. Example: Delete(8); Return 3 5

Deletion (Continued) If a sibling S of N has 2 items distribute items among N, S and the parent P; if N is internal, move the appropriate child from S to N. Else bring an item from P into S; If N is internal, make its (single) child the child of S; remove N. If P has no items Fix(P) (recursive call)

(2,4) Trees Size Property: nodes may have 1,2,3 items. Every node, except leaves has size+1 children. Depth property: all leaves have the same depth. Insertion: If during the search for the leaf you encounter a “full” node, split it.

(2,4) Tree

Insert(38);

Insert(105) Insert(105);

Removal As with BS trees, we may place the node to be removed in a leaf. If the leaf v has another item, done. If not, we have an UNDERFLOW. If a sibling of v has 2 or 3 items, transfer an item. If v has 2 or 3 siblings we perform a transfer

Removal If v has only one sibling with a single item we drop an item from the parent to the sibling, remove v. This may create an underflow at the parent. We “percolate” up the underflow. It may reach the root in which case the root will be discarded and the tree will “shrink”.

Delete(15)

Delete(15)

Continued Drop item from parent

Fuse

Drop item from root Remove root, return the child

Summary Both 2-3 tress and 2-4 trees make it very easy to maintain balance. Insertion and deletion easier for 2-4 tree. Cost is waste of space in each node. Also extra comparison inside each node. Does not “extend” binary trees.

Red-Black Trees Root property: Root is BLACK. External Property: Every external node is BLACK Internal property: Children of a RED node are BLACK. Depth property: All external nodes have the same BLACK depth.

RedBlack Insertion

Red Black Trees, Insertion 1. Find proper external node. 2.Insert and color node red. 3.No black depth violation but may violate the red-black parent-child relationship. 4.Let: z be the inserted node, v its parent and u its grandparent. If v is red then u must be black.

Color adjustments. Red child, red parent. Parent has a black sibling. a bu v w z Vl Zl Zr

Rotation Z-middle key. Black height does not change! No more red-red. a bz v w u Vl Zl Zr

Color adjustment II a bu v w z Vr Zl Zr

Rotation II a bv z u ZrZl w Vr

Recoloring Red child, red parent. Parent has a red sibling. a bu v w z Vl Zr

Recoloring Red-red may move up… a bu v w z Vl Zr Zl

Red Black Tree Insert 10 – root 10

Red Black Tree Insert 10 – root 10

Red Black Tree Insert

Red Black Tree Insert

Red Black Tree Rotate – Change colors

Red Black Tree Insert

Red Black Tree Change Color

Red Black Tree Insert

Red Black Tree Rotate – Change Color

Red Black Tree Insert

Red Black Tree Change Color

Red Black Tree Insert

Red Black Tree Rotate

Red Black Tree Insert

Red Black Tree Insert Oops, red-red. ROTATE!

Red Black Tree Double Rotate – Adjust colors Child-Parent-Gramps Middle goes to “top Previous top becomes child.

Red Black Tree Insert

Red Black Tree Insert

Red Black Tree Insert

Red Black Tree Insert

Red Black Tree Adjust color

Red Black Tree Insert

Red Black Tree Insert