### Examples

```Multi-Way search Trees
1. 2-3 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.
20 80
30 70
5
2 4
10
25
40 50 75
90 100
85
95 110 120
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);
20 80
5
2 3 4
10 15
30 70
23 25 40 50 75
90 100
85
95 110 120
Insert(3);
• In mid air…
20 80
5
30 70
90 100
3
2
4
10 15 23 25 40 50 75
85
95 110 120
Done….
20 80
3 5
2
30 70
4 10 15 23 25
40 50 75
90 100
85
95 110 120
Tree grows at the root…
• Insert(45);
20 80
3 5
2
4
30 70
10
25 40 45 50 75
90 100
85
95 110 120
• New root:
45
20
3 5
2
4
80
30
10
25 40
70
50
90 100
75
85
95 110 120
Delete
• If item is not in a leaf exchange with inorder 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.
45
20
3 5
2
4
85
30
10
25 40
70
50 75
90 100
80
95
110 120
• RedistributeRemove(80);
45
20
3 5
2
4
85
30
10
25 40
70
50 75
95 110
90
100
120
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 1 110 can spare an item: redistribute.
Delete(70)
45
20
3 5
2
4
85
30
10
25 40
75
50
95 110
90
100
120
New tree:
• Delete(75) will “shrink” the tree.
45
20
3 5
2
4
95
30
10
75
25 40 50
110
90
100
120
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
20 45
3 5
2
4
30
10
25 40
95 110
50 90
100
120
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);
5
5
1
3
2
8
3
3
Return
35
35
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
10
3 8
45
25
60
50 55
70 90 100
Insert(38);
Insert(38);
45
10
3 8
25 38
60
50 55
70 90 100
Insert(105)
• Insert(105);
45
10
3 8
25 38
60 90
50 55
70
100 105
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)
35
20
6
60
15
40 50
70 80 90
Delete(15)
35
20
6
60
40 50
70 80 90
Continued
• Drop item from parent
35
60
6 20
40 50
70 80 90
Fuse
35
60
6 20
40 50
70 80 90
Drop item from root
• Remove root, return the child.
35 60
6 20
40 50
70 80 90
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.
• Red child, red parent. Parent has a black
sibling.
a
b
u
w
v
z
Vl
Zl
Zr
Rotation
• Z-middle key. Black height does not
change! No more red-red.
a
b
z
u
v
Vl
Zl
Zr
w
a
b
u
w
v
Vr
z
Zl
Zr
Rotation II
a
b
v
u
z
Zl
Zr
Vr
w
Recoloring
• Red child, red parent. Parent has a red
sibling.
a
b
u
w
v
z
Vl
Zr
Recoloring
• Red-red may move up…
a
b
u
w
v
z
Vl
Zl
Zr
Red Black Tree
• Insert 10 – root
10
Red Black Tree
• Insert 10 – root
10
Red Black Tree
• Insert 85
10
85
Red Black Tree
• Insert 15
10
85
15
Red Black Tree
• Rotate – Change colors
15
10
85
Red Black Tree
• Insert 70
15
10
85
70
Red Black Tree
• Change Color
15
10
85
70
Red Black Tree
• Insert 20
15
10
85
70
20
Red Black Tree
• Rotate – Change Color
15
10
70
20
85
Red Black Tree
• Insert 60
15
10
70
85
20
60
Red Black Tree
• Change Color
15
10
70
85
20
60
Red Black Tree
• Insert 30
15
10
70
85
20
60
30
Red Black Tree
• Rotate
15
10
70
85
30
20
60
Red Black Tree
• Insert 50
15
10
70
85
30
20
60
50
Red Black Tree
• Insert 50
15
10
70
85
30
20
Oops, red-red.
ROTATE!
60
50
Red Black Tree
• Double Rotate – Adjust colors
30
15
10
Child-Parent-Gramps
Middle goes to “top
Previous top becomes
child.
70
20
60
50
85
Red Black Tree
• Insert 65
30
15
10
70
20
85
60
50
65
Red Black Tree
• Insert 80
30
15
10
70
20
85
60
50
65
80
Red Black Tree
• Insert 90
30
15
10
70
20
85
60
50
65
80
90
Red Black Tree
• Insert 40
30
15
10
70
20
85
60
50
40
65
80
90
Red Black Tree
30
15
10
70
20
85
60
50
40
65
80
90
Red Black Tree
• Insert 5
30
15
10
70
20
85
60
5
50
40
65
80
90
Red Black Tree
• Insert 55
30
15
10
70
20
85
60
5
50
40
65
55
80
90
```