Completely Balanced Trees

1
CompletelyBalanced Trees
2
Completely Balanced Trees

• So far, weve always grown our trees from the
  root to the leaf nodes
• Problem
• Unequal path lengths
• Goals
• Some maximum number of level traversals
• Expand from binary to N-ary trees
• For a tree containing N nodes, having M children
  per node where M grows large means that the
  height of the tree will be small

3
B-Trees

• Balanced trees
• Used for database index creation
• An on-disk data structure
• Nodes consist of
• N key values
• N data pointers (pointing to entire data item
  with that key, perhaps in a sequential file)
• N1 node pointers (pointing to other B-tree
  nodes)
• Note
• Can calculate N given size of pointers and keys,
  block size

4
B-Trees

• Each node may contain a large number of keys
• of subtrees of each node may be large
• An on-disk data structure
• Designed to branch out large number of directions
  
• Contain lots of keys in each node
• ? the height of the tree is relatively small N
  key values
• Small number of nodes must be read from disk to
  retrieve an item
• Large node size (with lots of keys in the node)
  but disk drive can usually read a fair amount of
  data at once.

5
B-Tree Definition

• A multi-way tree of order m is an ordered tree
  where each node has at most m children
• If k is the actual number of children in the
  node,
• ? k - 1 is the number of keys in the node.
• If the keys and subtrees are arranged in the
  fashion of a search tree, then this is called a
  multiway search tree of order m.

6
Multi-way Search Tree of order of 4
Keys
Pointers
7
B-Trees
8
B-Tree Properties

• A B-tree of order m is a multiway search tree of
  order m such that
• All leaves are on the bottom level.
• 2. All internal nodes (except perhaps the root
  node) have at least
• ?(m / 2)? (nonempty) children ? keep it bushy
  and balanced.
• 3. The root node can have as few as 2 children if
  it is an internal
• node, and can obviously have no children if the
  root node is a
• leaf (that is, the whole tree consists only of
  the root node).

9
B-Tree Properties
4. Each leaf node (other than the root node if
it is a leaf) must contain at least ?(m /
2)? - 1 keys. Note ?x? is the ceiling
function whose value is the smallest integer that
is greater than or equal to x. E.g., ?3?
3 ?3.34? 4 ?1.98 ? 2 ?5.001 ? 6
10
B-Tree Properties
A B-tree is a fairly well-balanced tree since all
leaf nodes must be at the bottom. Recall
condition 2. All internal nodes (except perhaps
the root node) have at least ?(m / 2)?
(nonempty) children ? keep it bushy and
balanced. Causes the tree to fan out, i.e.,
shorter height
11
B-Tree Insertion

• Insert keys in order in a single block until it
  fills
• When need to add value where there is no room,
  split the node into two nodes
• Smaller half (rounded down) of values go into
  first node
• Larger half (rounded down) of values go into
  second node
• Median value goes into new parent node
• Node pointers around median value point to two
  nodes at lower level

12
Insertion Example
Insert the following letters into what is
originally an empty B-tree of order 5 C N G A
H E K Q M F W L T Z D P R X Y S Order 5 ? max
of 5 children and 4 keys. All nodes (except
root) must have a minimum of 2 keys. Inserting
in alphabetical order the first 4 letters
13
Insertion Example
Insert H next. No room. Split into 2 nodes.
Move median item G up into new root node
Insert EKQ next
14
Insertion Example
C N G A H E K Q M F W L T Z D P R X Y
S Inserting E, K, Q doesnt require splits.
But inserting M does
?? - split into 2
15
Insertion Example
C N G A H E K Q M F W L T Z D P R X Y S F, W,
L, and T are then added without needing any
split.
16
Insertion Example
C N G A H E K Q M F W L T Z D P R X Y S F, W,
L, and T are then added without needing any
split.
Move median (T) up split node
Adding Z requires node to split
17
Insertion Example
Z
18
C N G A H E K Q M F W L T Z D P R X Y S
D (which is the median too)
Insert PRXY without any splitting
19
C N G A H E K Q M F W L T Z D P R X Y S

• - When S is inserted, node must split ? Q
  (median) goes up.
• Qs parent is full, split further ? make M
  (medium of Qs parent)
• go up.

20
B Tree Deletion

• Delete H
• H is a leaf node. Easy.
• Move K and L over to the left

21
B Tree Deletion

• Delete T (non-leaf) Find Ts successor (i.e., W)
  and move it up to replace T

22
B Tree Deletion

• Delete T
• In all cases, delete leaf if leaf has extra keys

23
B Tree Deletion

• Delete R next
• R is a leaf node.
• Move S to Rs spot, move W to Ss spot
• Move X up to Ws spot

?
ß
a
24
B Tree Deletion
25
B Tree Deletion
- Delete E next - Very problematic ? siblings as
well as E has no extra keys - Combine the leaf
with one of two siblings - Move down parents key
that was between these two siblings
26
B Tree (Delete E)
27
B Tree Deletion

• - But node G must have at least ?(5 / 2)? -1
  keys
• G cannot borrow key from sibling (QX node) (no
  extra keys)
• Combine siblings and parent node into one (root)
  node

28
B Tree Deletion
29
B Trees
30
B-Tree

• Problem with B-Tree
• Somewhat unequal access times
• Difficult to traverse index sequentially
• B-Tree
• All data stored at lowest level (leaf nodes)
• Some values duplicated in internal nodes for
  indexing
• Cost extra storage, duplication, two types of
  nodes
• Benefits sequential access across bottom level

31
B-Trees

• Variant of B-trees
• All data stored at lowest level (leaf nodes)
• All leaves are linked sequentially
• Used as a dynamic indexing method in relational
  DBs
• Contains index pages and data pages.
• Root and non-leaf nodes are index pages.

32
B-Tree Example

• With 4 keys
• Leaf nodes are linked to each other via doubly
  linked lists (not shown)

33
B-Tree Insertions

• Key value determines placement
• Three cases for insertion

34
(No Transcript)
35
(No Transcript)
36
Insertion Example
Adding record with Key 28
37
Insertion Example

• Adding record with Key 70
• Should go into leaf containing 50 55 60 65 -gt
  too bad, its full
• Split page as follows
• Left leaf Right leaf
• 50 55 60 65 70
• 3. Middle key(60) placed in the corresponding
  parent index page

38
Insert (Leaf full index not)
70
39
Insert (Leaf and index pages are full)
Add 95 ? belongs here Split leaf page into 2
75 80 and 85 90 95 Middle key (85)
goes up ? parent is full ? split parent 25 50
60 75 85 Middle key (60) made a
new parent of the parents
40
Insert (Leaf and index pages are full)
95
41
Rotation
When a leaf node is full and its sibling is
not. Reduce number of page splits. E.g., add 70 ?
previously, we split the 50 55 60 65 node and
brought 60 up Instead, move a record to its
sibling
42
(No Transcript)
43
Deletion Example
Delete 70 OK since fill factor 50 (min
records in a node)
44
Deletion Example
Now delete 25 Leaf OK (fill factor
satisfied) Index not OK ? replace with 28
45
Deletion Example
Delete 60 fill factor lt 50 ? combine leaf
pages and index pages
46
Deletion Example