Binary Trees

1
Binary Trees

• A binary tree is made up of a finite set of nodes
  that is either empty or consists of a node called
  the root together with two binary trees, called
  the left and right subtrees, which are disjoint
  from each other and from the root.

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Binary Tree Example

• Definitions
• Node element of the tree
• Root top node
• Subtree binary tree associated with the root
  node (left and right)
• Children the root node of a nodes subtrees
• Edge a link from a node to its children
• Parent the node to which the current node is a
  child
• Ancestor, descendant

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Binary Tree Example

• Path a sequence of nodes n1,n2,,nk s.t. ni is
  a parent of ni1 for 1 lt i lt k
• Length of a path k -1
• Depth length of path from root to a node
• Height depth of the deepest node 1
• Level all nodes of depth d
• Leaf node with two empty children
• Internal node node with at least one non-empty
  child

No standard definitions
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Full and Complete Binary Trees

• Full binary tree Each node is either a leaf or
  internal node with exactly two non-empty
  children.
• Complete binary tree If the height of the tree
  is d, then all leaves except possibly level d are
  completely full. The bottom level has all nodes
  to the left side.

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Full Binary Tree Theorem (1)

• Theorem The number of leaves in a non-empty full
  binary tree is one more than the number of
  internal nodes.
• Proof (by Mathematical Induction)
• Base cases A full, non-empty binary tree with 0
  internal nodes must have 1 leaf node. A full
  binary tree with 1 internal node must have two
  leaf nodes.
• Induction Hypothesis Assume any full binary tree
  T containing n internal nodes has n1 leaves.

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Full Binary Tree Theorem (2)

• Induction Step Given tree T with n internal
  nodes, pick leaf node L and make it internal (by
  adding two children). T is still full. How many
  internal leaf nodes does it contain?
• We added two new leaf nodes, but lost one
• n 1 2 1 n 2
• We created one new internal node
• n 1
• T has n1 internal nodes, and n2 leaf nodes, or
  n internal nodes and n1 leaf nodes
• By induction, the theorem holds for n gt 0

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Traversals (1)

• Any process for visiting the nodes in some order
  is called a traversal.
• Any traversal that lists every node in the tree
  exactly once is called an enumeration of the
  trees nodes.

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Traversals (2)

• Preorder traversal Visit each node before
  visiting its children.
• Postorder traversal Visit each node after
  visiting its children.
• Inorder traversal Visit the left subtree, then
  the node, then the right subtree.
• Level-order traversal Visit all nodes in level
  0, then level 1, then level 2, etc.
• Always visiting the left child first

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Recursive Implementation

• Pre-, post-, and in-order traversals are all
  easily written as recursive functions (or with a
  stack)
• Level-order requires an additional data structure
  (a queue)

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What are the traversals?
Give the pre-, post-, in-, and level order
traversal of the tree.
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Counting Nodes

• We can count the number of nodes in a tree using
  a traversal.
• Key point the number of nodes in a tree with
  root node r is the number of nodes in rs left
  subtree plus the number of nodes in rs right
  subtree plus 1 (for r)

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Array Implementation
13
Array Implementation

• Parent(r) (r-1)/2 if r ltgt 0 and r lt n.
• Leftchild(r) 2r 1 if 2r 1 lt n.
• Rightchild(r) 2r 2 if 2r 2 lt n.
• Leftsibling(r) r - 1 if r is even, r gt 0, and r
  lt n.
• Rightsibling(r) r 1 if r is odd and r 1 lt
  n.

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Binary Tree Implementation
15
Array Implementation
16
Array Implementation

• Parent(r) (r-1)/2 if r ltgt 0 and r lt n.
• Leftchild(r) 2r 1 if 2r 1 lt n.
• Rightchild(r) 2r 2 if 2r 2 lt n.
• Leftsibling(r) r - 1 if r is even, r gt 0, and r
  lt n.
• Rightsibling(r) r 1 if r is odd and r 1 lt
  n.

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Binary Search Trees

• BST Property All elements stored in the left
  subtree of a node with value K have values lt K.
  All elements stored in the right subtree of a
  node with value K have values gt K.

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BST Search

• Algorithm
• search(K,T) for key K and tree T
• if currnode is NULL, return false
• if K lt currnode-gtvalue
• return search(K,currnode-gtleft)
• else if K gt currnode-gtvalue
• return search(K,currnode-gtright)
• else // must equal currnode
• return true

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BST Insert (1)
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BST Insert (2)

• Insert algorithm (no duplicates)
• insert(K,T) insert key K into Tree T
• Find leaf node L where search(K,T) fails
• If K lt L-gtvalue
• add K as Ls left child
• else
• add K as Ls right child

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BST Remove (1)
22
BST Remove

• Remove algorithm
• Remove(K,T) remove key K from tree T
• find node N with K using search(K,T)
• if N is a leaf, remove and exit
• else
• find the smallest node in the right subtree
  of N, called n N
• exchange values for N and N
• remove N

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Cost of BST Operations

• Find
• Insert
• Delete
• All cost depth of the node in question.
• Worst case ?(n).
• Average case ?(log n).

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Heaps

• Heap Complete binary tree with the heap
  property
• Min-heap All values less than child values.
• Max-heap All values greater than child values.
• The values are partially ordered.
• Heap representation Normally the array-based
  complete binary tree representation.

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Building the Heap

• (a) (4-2) (4-1) (2-1) (5-2) (5-4) (6-3) (6-5)
  (7-5) (7-6)
• (b) (5-2), (7-3), (7-1), (6-1)

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Siftdown (1)

• For fast heap construction
• Work from high end of array to low end.
• Call siftdown for each item.
• Dont need to call siftdown on leaf nodes.
• Siftdown(H)
• if(H-gtvalue gt H-gtleft-gtvalue and
• H gt H-gtright-gtvalue) then return
• else
• H max(H-gtleft, H-gtright)
• swap values(H,H)
• siftdown(H)

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Siftdown (2)
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Buildheap Cost

• Cost for heap construction
• log n
• ? (i - 1) n/2i ? n.
• i1

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Remove Max Value

• RemoveMax(H) -- get the max element off heap H
• maxelem H-gtvalue // Max is at the top
• H-gtvalue last elem-gtvalue
• shiftdown(H)
• return maxelem

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Priority Queues (1)

• A priority queue stores objects, and on request
  releases the object with greatest value.
• Example Scheduling jobs in a multi-tasking
  operating system.
• The priority of a job may change, requiring some
  reordering of the jobs.
• Implementation Use a heap to store the priority
  queue.

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Priority Queues (2)

• To support priority reordering, delete and
  re-insert. Need to know index for the object in
  question.

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Huffman Coding Trees

• ASCII codes 8 bits per character.
• Fixed-length coding.
• Can take advantage of relative frequency of
  letters to save space.
• Variable-length coding
• Build the tree with minimum external path weight.

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Huffman Tree Construction (1)
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Huffman Tree Construction (2)
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Assigning Codes
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Coding and Decoding

• A set of codes is said to meet the prefix
  property if no code in the set is the prefix of
  another.
• Code for DEED
• Decode 1011001110111101
• Expected cost per letter