Chapter 6. Trees II

6.3 Binary Trees

A binary tree is an ordered tree in which every node has at most two children. A binary tree is proper if each internal node has two children.

  • A binary tree is a tree with the following properties:
    • Each internal node has two children
    • The children of a node are an ordered pair
  • We call the children of an internal node left child and right child
  • Alternative recursive definition: a binary tree is either
    • a tree consisting of a single node, or
    • a tree whose root has an ordered pair of children, each of which is a binary tree

Examples: arithmetic expression, decision tree

Arithmetic Expression Tree

  • Binary tree associated with an arithmetic expression
    • internal nodes: operators
    • external nodes: operands
  • Example: arithmetic expression tree for the expression
    (2 Ч (a −1) + (3 Ч b))

Decision Tree

6.3.1 Binary Tree ADT

6.3.2 A Binary Tree Interface

html-6.14a (InspectableBinaryTree)
html-6.14b (BinaryTree)

6.3.3 Properties of Binary Trees

We denote the set of all nodes of a binary tree T, at the same depth d, as the level d of T.
Proposition 6.9: Let T be a proper binary tree with n nodes and let h denote the height of T. Then T has the following properties:
  1. The number of external nodes e in T is: h + 1 e 2h
  2. The number of internal nodes i in T is: h i 2h 1
  3. The total number of nodes n in T is: 2h + 1 n 2h+1 1
  4. The height h of T is: log(n + 1) 1 h (n 1)/2
Proposition 6.10: In a proper binary tree  T, the number of external nodes e is 1 more than the number of internal nodes i, i.e. e = i + 1.

  • Notation
    n number of nodes
    e number of external nodes
    i number of internal nodes
    h height

  • Properties:
    • e = i + 1
    • n = 2e − 1
    • h ≤ i
    • h ≤ (n − 1)/2
    • e ≤ 2h
    • h ≥ log2e
    • h ≥ log2(n + 1)−1

Traversals of a Binary Tree

Preorder Traversal of a Binary Tree
void binaryPreorderPrint(const Tree& T, const Position& v) 
{ cout << v.element(); // print element
if (isInternal(v)) // visit children
{ cout << " ";
binaryPreorderPrint(T, T.leftChild(v));
binaryPreorderPrint(T, T.rightChild(v));
Postorder Traversal of a Binary Tree
void binaryPostorderPrint(const Tree& T, const Position& v) 
{ if (isInternal(v)) // visit children
{ cout << " ";
binaryPostorderPrint(T, T.leftChild(v));
binaryPostorderPrint(T, T.rightChild(v));
cout << v.element(); // print element
Evaluating an Arithmetic Expression

  • Specialization of a postorder traversal
    • recursive method returning the value of a subtree
    • when visiting an internal node, combine the values of the subtrees
  • O(n) time algorithm

Algorithm evalExpr(v)
      if isExternal (v)
            return v.element ()
          xevalExpr(leftChild (v))
          yevalExpr(rightChild (v))
          ◊ ← operator stored at v
          return xy

Inorder Traversal of a Binary Tree

  • In an inorder traversal a node is visited after its left subtree and before its right subtree
  • Application: draw a binary tree
    • x(v) = inorder rank of v
    • y(v) = depth of v
  • Visit "left to right"

Algorithm inOrder(v)
   if isInternal (v)
       inOrder(leftChild (v))
   if isInternal (v)
      inOrder(rightChild (v))

void binaryInorderPrint(const Tree& T, const Position& v) 
{ if (isInternal(v)) // visit left child
binaryInorderPrint(T, T.leftChild(v));
cout << v.element(); // print element
if (isInternal(v)) // visit right child
binaryInorderPrint(T, T.rightChild(v));
Binary Search Trees

    Binary search tree
is a binary tree so that each internal node v stores an element e, such that:
Position searchBinaryTree(const Tree& T, const Position& v, const Object& e)
{ if (isInternal(v))
if (v.element() == e) return v; // found!
else if (v.element() < e)
searchBinaryTree(T, T.leftChild(v), e);
// search left subtree
searchBinaryTree(T, T.rightChild(v), e); // search right subtree
else return ... // not found!
The time for searching in a binary tree T is proportional to the height of T, i.e. >= O(log n) and <= Omega(n)

A Unified Tree Traversal Framework

The Euler Tour Traversal of a Binary Tree

  • Generic traversal of a binary tree
  • Includes a special cases the preorder, postorder and inorder traversals
  • Walk around the tree and visit each node three times:
    • on the left (preorder)
    • from below (inorder)
    • on the right (postorder)

6.4 Data Structures for Representing Trees

6.4.1 A Vector-Based Structure for Binary Trees

6.4.2 A Linked Structure for Binary Trees

Nodes and Positions in a Binary Tree
html-6.27 (Node)
html-6.28 (Position)

Binary Tree Update Functions
html-6.29 (LinkedBinaryTree1)
html-6.30 (LinkedBinaryTree2)


6.4.3 A Linked Structure for General Trees

6.4.4 Representing General Trees with Binary Trees

A representation of a general (ordered) tree T  is obtained by transforming T  into a binary tree T '. The transformation is as follows:

Additional links: