General Trees and Variants CPSC 335

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General Treesand VariantsCPSC 335
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Lecture Outline

• General Treesand transformation to binary trees
• B-tree variants B, B, prefix B
• 2-4, Horizontal-vertical, Red-black trees

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General Trees

• Trees in which the number of subtrees of any
  node need not be required to be 0, 1, or 2.
• The number of subtrees for any node may be
  variable.
• Some nodes may have 1 or no subtrees, others
  may
• have 3, some 4, or any other combination.
• Ternary tree 3 subtrees per node

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Problems with General Trees and Solutions

• Problems
• The number of references for each node must
  be equal to the maximum that will be used in the
  tree.
• Most of the algorithms for searching,
  traversing, adding and
• deleting nodes become much more complex in that
  they must now cope with situations where there
  are not just two possibilities for any node but
  multiple possibilities.
• Solution
• General trees can be converted to binary
  trees
• The algorithms that are used for binary tree
  processing can be used with minor modifications.

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Creating a Binary Tree from a General Tree

• Conversion process
• Use the root of the general tree as the root
  of the binary tree.
• Determine the first child of the root. This
  is the leftmost node in the general tree at the
  next level.
• Insert this node. The child reference of
  the parent node refers to this node.
• Continue finding the first child of each
  parent node and insert it below the parent node
  with the child reference of the parent to this
  node.

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Creating a Binary Tree from a General Tree

• Conversion process (continued)
• When no more first children exist in the
  path just used, move back to the parent of the
  last node entered and repeat the above process.
• Complete the tree for all nodes. In order to
  locate where the node fits you must search for
  the first child at that level and then follow the
  sibling references to a nil where the next
  sibling can be inserted. The children of any
  sibling node can be inserted by locating the
  parent and then inserting the first child. Then
  the above process is repeated.

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Creating a Binary Tree from a General Tree

General Tree Binary Tree
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Traversing a Tree

• When the general tree has been represented as
  a binary tree, the algorithms which were used for
  the binary tree can now be used for the general
  tree.
• In-order traversals make no sense when a
  general tree is converted to a binary tree. In
  the general tree each node can have more than two
  children so trying to insert the parent node in
  between the children is rather difficult,
  especially if there is an odd number of
  children.
• Pre - order
• This is a process where the root is
  accessed and processed and then each of the
  subtrees is preorder processed. It is also
  called a depth-first traversal.

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Traversing a Tree

• Pre-order Traversal
• In this way the resulting printout has all
  nodes at any given level starting in the same tab
  column.
• It is relatively easy to draw lines to
  produce the original general tree except that the
  tree is on its side with it's root at the left
  rather than with the root at the top.

Result of pre-order traversal
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Part 2

• General Treesand transformation to binary trees
• B-tree variants B, B, prefix B
• 2-4, Horizontal-vertical, Red-black trees

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B-tree variants B

• B - introduced by D. Knuth
• A B-tree variant where all nodes, except the
  root, are required to be at least 2/3 full
• The implementation is more complex, since instead
  of combining two nodes into one on deletes, and
  splitting one node into two on inserts, you have
  to combine three nodes into two, and split two
  nodes into three
• A split operation is however delayed by an
  attempt to redistribute keys between neighbours
  (similarly to a B tree deletion) when the node
  becomes full. So, instead of a split, you first
  need to try to redistribute keys in the node, its
  neighbor and the key in the parent node.

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B-tree variants B

• B tree is a B tree where data is stored only in
  the leaves, never higher up. This makes for more
  compact parent nodes (which now contain only
  keys), at the cost of some redundancy the keys
  in the parents are duplicated in the leaves.
• Note that this changes the deletion of the keys
  in the parents, in that the nodes to the right of
  a key are now greater than or equal to that key.
  This complicates the implementation somewhat, as
  leaf nodes are now different than the other
  nodes.

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B-tree variants B

• Each leaf can hold up to n 1 values and contain
  at least (n 1) / 2 values.
• Nonleaf node pointers point to tree nodes (leaf
  nodes). Nonleaf nodes can hold up to n pointers
  and must hold at least n/2 pointers.

PE500
MA244
RA200
BT100 DD112
MA244
RA200 RA687
PE500
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B-tree variants B

• Insertion if the new node has a search key that
  already exists in another leaf node, then adds
  the new record to the file and a pointer to the
  bucket of pointers. If the search key is
  different from all others, it is inserted in
  order.
• Deletion remove the search key value from the
  node.

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Prefix B tree

• Prefix B tree (Bayer, Unterauer) is a B tree
  where index uses the SHORTEST possible separators
  needed to distinguish two keys.
• Full key is not stored thus instead of storing
  CD244 in the node, only C or CD or CD2 will be
  stored depending onother values in the tree.

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Bit tree

• Bit trees introduced by Ferguson in 1999
  represented keys as bits and stored only the
  first portion of the bit string (similarly to
  extendible hashing) that was required to separate
  2 keys.

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Summary on B-trees and variants

• B (D. Knuth) is a B-tree variant where all
  nodes, except the root, are required to be at
  least 2/3 full and split operation is delayed
• B tree is a B tree where data is stored only in
  the leaves, and keys are duplicated in the index.
• Prefix B tree (Bayer, Unterauer) is a B tree
  where index uses the SHORTEST possible separators
  needed to distinguish two keys.
• Bit trees (Ferguson,1999) represent keys as bits
  and stored only the first portion of the bit
  string that is necessary to separate 2 keys.

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Part 3

• General Treesand transformation to binary trees
• B-tree variants B, B, prefix B
• 2-4, Horizontal-vertical, Red-black trees

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2-4 trees, horizontal-vertical, red-black

• 2-4 trees are B trees of order 4, i.e. with 3
  keys in the node.
• They can be easily represented as a binary tree
  and have the following advantages
• Insertions are easily and faster handled
• Trees are balanced
• Easily transformed back and forth from B tree to
  binary tree

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2-4 trees, horizontal-vertical, red-black

• Intermediate result of transformation is called
  the horizontal-vertical tree
• Each node is a single key node that has pointers
  (horizontal) onto its former neighbors (keys from
  the same node) from the 2-4 tree or vertical onto
  former children

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B-tree variants from horizontal-vertical to
binary tree
3
8
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1
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• The path from root to any node in
  horizontal-vertical tree contains the same number
  of vertical links

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Split operation in 2-4 tree vs. horizontal
vertical

• In order to perform a split (when inserting) in
  2-4 tree, a new node needs to be created and two
  pointers adjusted, plus keys moved.
• In horizontal-vertical tree all that is required
  is to change the type of link from horizontal to
  vertical! This is called bit flipping operation
  and it is extremely fast!

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Red-black trees

• Red-black tree are a variation of the BST
  (similar to above) that are balanced thus the
  path from the root of the tree to the leaf
  contains the same number of black nodes
• Root is black
• Each node contains 1 bit (color) black or red
• If node is red, than both children are black
• Every red node is either
• A full node (with two black children) or
• A leaf node
• On any path from the root to a leaf, red nodes
  must not be adjacent. However, any number of
  black nodes may appear in a sequence.

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Red-black trees
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Red-black properties

• Height is O(lgn), n number of nodes
• All operations can be done in O(lgn time)
• Search is the same as in BST
• Insertion/deletion involve rotations to keep tree
  balanced (similarly to AVL tree).

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Red-black properties

• Red-black trees offer worst-case guarantees for
  insertion time, deletion time, and search time.
• Used in time-sensitive applications such as
  real-time applications, and also in algorithms
  which provide worst-case guarantees
  (computational geometry)
• The AVL tree is another structure supporting
  O(log n) search, insertion, and removal. AVL tree
  is more rigidly balanced than Red-Black trees,
  leading to slower insertion and removal but
  faster retrieval as AVL tree is more compact.

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Red-black tree worst-case
Red-black properties
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Red-black properties
Douglas Wilhelm Harder, UWaterloo
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Summary on 2-4 trees and variants

• 2-4 trees are B trees of order 4, i.e. with 3
  keys in the node.
• They can be easily represented as a binary tree
  and are balanced, with easy insertion operations
• Intermediate result of transformation from 2-4
  tree to binary tree is called the
  horizontal-vertical tree
• Red-black tree are a variation of the BST that
  are balanced, the path from the root of the tree
  to the leaf contains the same number of black
  nodes. They support O(log n) search, insertion,
  and removal and is less rigidly balanced than AVL
  tree

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Conclusions

• The use of the discussed variants of the trees
  depends greately on the conditions of the given
  problem and needed features.
• While B trees and their variants used for
  indexing of databases, GIS systems, and file
  accesses, the 2-4, balanced and red-black trees
  are used instead of BST for fast information
  retrieval, in games, computer graphics, and
  computer modeling.