Binary Trees

1
Binary Trees

• A binary tree is a tree with the following
  properties
• Each internal node has at most two children
• The children of a node are ordered
• Left Child
• Right Child
• Applications
• arithmetic expressions
• decision processes
• Searching (Binary Search Trees)
• Heap implementation of a Priority Queue

2
Binary Trees(Recursive Definition)

• A binary tree is either empty or consists of
• A node r, called the root of tree T
• A binary tree (Tleft) called the left subtree of
  T
• A binary tree (Tright) called the right subtree
  of T

3
Decision Tree

• Binary tree associated with a decision process
• internal nodes questions with yes/no answer
• external nodes decisions

4
Arithmetic Expression Tree

• Binary tree associated with an arithmetic
  expression
• external nodes operands
• Value is that of its variable or constant
• internal nodes operators
• Value is defined by applying its operation to the
  value of its children
• Example arithmetic expression tree for the
  expression
• (2 ? (a - 1) (3 ? b))

5
Binary Trees

• A binary tree T of height h is Full if
• All of the leaves are at level h and every parent
  (internal node) has exactly two children.

Full
Not Full
6
Binary Trees

• A binary tree T is Proper if
• All nodes have zero or two children.

Proper
Improper
7
Binary Trees

• A binary tree T is Complete if
• All levels of the binary tree, except the last,
  contain as many nodes as possible, and the nodes
  on the last level are filled from left to right.

8
Properties of Full Binary Trees

• The number of nodes at depth d is 2d
• The height of the tree is log(n1)-1
• Number of external nodes equals number of
  internal plus 1 (ne ni 1)
• Additional properties in section 7.3.3

Depth 0
Depth 1
Depth 2
Depth 3
9
BinaryTree ADT

• The BinaryTree ADT extends the Tree ADT, i.e., it
  inherits all the methods of the Tree ADT
• Accessor methods
• left(v)
• right(v)
• Query methods
• hasLeft(v)
• hasRight(v)
• Update
• addRoot(e)
• insertLeft(v, e)
• insertRigth(v, e)
• attach(v, T1, T2)
• remove(v)

10
Tree ADT

• Accessor methods
• root()
• parent(v)
• children(v)
• Query methods
• isInternal(v)
• isExternal(v)
• isRoot(v)
• Update method
• replace (v, e)
• Genereral methods
• size()
• isEmpty()
• elements()
• nodes()

11
Inorder Traversal

• In an inorder traversal a node is visited after
  its left subtree and before its right subtree

Algorithm inOrder(v) if hasLeft (v) inOrder (left
(v)) visit(v) if hasRight (v) inOrder (right (v))
6
2
8
1
7
9
4
3
5
Demo
12
Print Arithmetic Expressions

• Inorder traversal
• print operand or operator when visiting node
• print ( before traversing left subtree
• print ) after traversing right subtree

Algorithm printExpression(v) if
hasLeft(v) print(() inOrder (left(v)) print(v.e
lement()) if hasRight(v) inOrder(right(v)) print(
))
((2 ? (a - 1)) (3 ? b))
13
Inorder Traversals Compared
Arithmetic Expressions
General
Algorithm inOrder(v) if hasLeft(v) inOrder(left(v)
) visit(v) if hasRight(v) inOrder(right(v))
Algorithm printExpression(v) if hasLeft
(v) print(() printExpression(left(v)) print(v.e
lement()) if hasRight(v) printExpression(right(v))
print())
14
Evaluate Arithmetic Expressions

• Postorder traversal
• recursive method returning the value of a subtree
• when visiting an internal node, combine the
  values of the subtrees

Algorithm evalExpr(v) if isExternal(v) return
v.element() else x ? evalExpr(leftChild(v)) y ?
evalExpr(rightChild(v)) ? ? operator stored at
v return x ? y
15
Linked Structure for Binary Trees

• Binary Tree Node
• Element
• Parent node
• Left child node
• Right child node

Parent
Left
Right
Element
16
An Array Structure for Binary Trees
Given node x left(x) 2x right(x)
2x1 parent(x) floor(x/2)
17
An Array Structure for Binary Trees(Example)
1
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15
a b d h
l n o
18
Array vs. Linked Tree Implementation

• Storage requirements

• Worst case array of size 2n required to store a
  tree of size n

• Run time of basic binary tree operations

• Basically the same, but what about inserting an
  element?