1. TCSS 342, Winter 2006 Lecture Notes
Multiway Search Trees
version 1.0

2. Objectives Define Multiway Search trees
Show through how trees are kept balanced
Show the connection between trees and red-black trees
Explain the motivation behind B-Trees.

3. Multi-Way search Trees
Like Binary Search trees except:
nodes can have more than one element
perfectly balanced (all leaf nodes at same level)
Still obeys ordering property
(2,3) tree
contains only 2-nodes and 3-nodes
2-node: one item node with 0 or 2 children
same ordering property as binary search tree nodes
3-node: two item node with 0 or 3 children
With two items k1 and k2 and 3 children c0, c1, c2, ordering property is: c0 < k1 < c1 < k2 < c2.
all leaves at same level.

4. Example A (2,3) tree storing 18 items. 20 80 5 30 70 90 100 25 40 50
20 80 5
30 70 25
40 50 75 85
2 4 10 95

5. Multiway search trees (2,4) tree Alternative 2-4 tree definition:
contains only 2-nodes, 3-nodes, and 4-nodes
4-node: three item node with 0 or 4 children; satisfies similar ordering property
all leaves at same level
Alternative 2-4 tree definition:
All nodes must contain between 1 and 3 items
All leaf nodes must be at the same level
Every internal (non-leaf) node satisfies:
If it contains i items k1, k2, .. ki, then
It must have i+1 children c0, c1, .. ci
Items are sorted so that for each item kb,
All items in subtree cb-1 are less than kb.
All items in subtree cb are greater than kb.

6. (2,4) Tree Example
3 8
50 55 25

7. B-Trees (2,3) Tree = B-tree of order 3 (2,4) Tree = B-tree of order 4
In general, B-tree of order m
allows k items per node, where
m/2 -1  k  m-1
Internal nodes with k items have k+1 children
Satisfies similar ordering property
All leaves at same level
Design ideas behind B-Trees:
In array implementation, each node at most ½ empty
(# items per node allowed not strictly followed when there are not enough items)
Always balanced

8. B-Trees motivation Suppose data stored on disk
To read data, speed depends on
Seek time (moving head)
Transfer time (after location is found on disk)
Accessing a contiguous chunk of data relatively fast
Accessing data scattered in different parts of disk slow
Storing sorted data on disk:
Use B-tree to store long contiguous chunks
Design order of B-tree so items in one node fits in exactly one fast-loading chunk of data.

9. Implementing multiway search trees
Always maintain the search tree properties after every insertion and deletion
Maintain ordering property
Maintain all leaves at same level
Make sure #items in each node is acceptable
Do some combination of the following to maintain tree properties:
Rotations (Redistributions)
Splitting full nodes
Fusing neighboring nodes together
Moving item from child to parent, or vice versa
Replacing an item with its in-order successor.
Do case-by-case analysis to figure out exact steps

10. (2,4) Trees, some details While adjusting trees Insertion:
Temporarily allow 4 items in a node.
Temporarily allow leaves to not be at same level.
Insertion:
Find and insert item into proper leaf node (by sorted order)
If node is too full (has 4 items), then split the node
Split full node into one node with two subtrees
Leaves of new subtrees are now one level off
Merge root of split node upward with parent
Fixes level problem
Repeatedly apply split to parent node, if necessary.
If process goes all the way top, may get new root.
Alternative insertion strategy: split full nodes on the way down when looking for proper leaf node.

11. (2,4) Tree
Next: Insert 38
3 8
50 55 25

12. After insert(38); next: insert(105)
(2,4) Tree
After insert(38); next: insert(105) 45 10 60
3 8
25 38
50 55

13. (2,4) Tree
After Insert(105); 45 10
60 90
3 8
50 55 70
25 38

14. Removal Swap node to removed with inorder successor
Inorder successor will always be in a leaf.
Remove item from the leaf node.
If node is now empty, try to fill it with an extra item from neighbor (redistribution/rotation).
If all neighbors have only one item, we have an UNDERFLOW. Fix this by
move parent item down into child node so it contains two items
move a grandparent item down into parent node
merge parent node with sibling (possible node split)
If necessary, fix UNDERFLOW at grandparent level
Underflows propagating to root shrink tree.

15. Delete(15) 35 20 60 6 15
40 50

16. Delete(15) 35 20 60 6
40 50

17. Continued
Drop item from parent 35 60
6 20
40 50

18. Continued
Drop item from grandparent 35 60
6 20
40 50

19. Fuse
35 60
6 20
40 50

20. Drop item from root Remove root, return the child. 35 60 6 20 40 50
35 60
6 20
40 50

21. Summary (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.

22. Red-Black and (2,4) trees.
Every Red-Black tree has a corresponding (2,4) tree.
Move every red node upwards into parent black node.
Does this always work? Why?
max # items in a node?
all leaves at same depth?
Is there only one unique way to convert
red-black tree into (2,4) tree?
(2,4) tree into red-black tree?
Strategy for red-black deletion:
Convert Red-Black to (2,4) tree.
Delete from (2,4) tree.
Convert back to red-black tree.

24. The corresponding (2,4) tree
10 30 20 5
Deleting 20 is a simple transfer!
Remove the item 20. Drop an item (30) from parent.
Move an item 40  parent.

25. Updated tree
10 40 5 30
50 55

26. Now redraw the RBTree! 60 40 70 10 85 50 65 5 30 80 90 55
Drill: delete(65), delete(80), delete(90), delete(85)

27. References
Lewis & Chase book, chapter 16.

28. Example A (2,3) tree storing 18 items. 20 80 5 30 70 90 100 25 40 50
20 80 5
30 70 25
40 50 75 85
2 4 10 95

29. 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 items among two children.
Insert(3);

30. Insertion
Insert(3);
20 80 5
30 70
40 50 75 85
10 15
23 25 95

31. Insert(3);
In mid air…
20 80 5
30 70 3
23 25
40 50 75 85 2
10 15 95 4

32. Done….
20 80
3 5
30 70 4
23 25
40 50 75 85 2
10 15 95

33. Tree grows at the root… Insert(45); 20 80 3 5 30 70 90 100 4 25
20 80
3 5
30 70 4 25 75 85 2 10 95

34. New root: 45 20 80
3 5 30 70 4 25 40 50 75 85 2 10 95

35. 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);

36. Remove(80); Step 1: Exchange 80 with in-order successor. 45 20 85 3 5
3 5 30 70 4 25 40 50 75 80 2 10 95

37. Redistribute Remove(80); 45 20 85 3 5 30 70 95 110 120 4 25 40 50 75
3 5 30 70
120 4 25 40 50 75 90 2 10
100

38. Some more removals… Remove(70); Swap(70, 75);
“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.
110

39. Delete(70) 45 20 85
3 5 30 75
120 4 25 40 50 90 2 10
100

40. New tree: Delete(85) will “shrink” the tree. 45 20 95 3 5 30 85 110
3 5 30 85
110
120 4 25 40 50 90
100 2 10

41. Details 1. Swap(85, 90) //inorder successor
2. Remove(85) //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.

42. “Shorter” 2-3 Tree
20 45
3 5 30
95 110
50 90
120 4 25 40
100 2 10

43. 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.

44. Deletion (continued)
If N is the root, delete it and return its child as the new root.
Example: Delete(8); 5 5 1 2 3 3
3 5 3 8
Return
3 5

45. 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)