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TREES
The function of education is to teach one to
think intensively and to think critically.
Intelligence plus character - that is the goal of
true education.
Martin Luther King, Jr.
Smitha N Pai
• Basic terminologies
• Binary tree representation
• Recursive/ non recursive inorder, preorder and post order tree
traversal
• Binary search tree
• AVL trees.
• Applications:
• Expression trees
• Inserting, deleting, searching, height of BST
8 hrs
• Ellis Horowitz, Sartaj Sahni , Anderson, “ Fundamentals of Data
Structures in C”, Silicon Press, 2 nd Edition, 2007.
Tree terminology
• A tree is a finite set of one or more nodes such that
• There is a specially designed node called the root
• The remaining nodes are partitioned into n>=0 disjoint sets T1, ….Tn, where
each of these sets is a tree. We call T1…Tn the subtrees of the root
• A node stands for the item of information and the branches to
other nodes
• Degree of a node is the number of subtrees of the node
• Degree of a tree is the maximum degree of the nodes in the tree
• A node that has subtrees is the parent of the roots of the subtrees
and the roots of the subtrees are the children of the node
• Indegree
• Outdegree
• Isolated node
• Directed tree –one node with indegree 0 and other nodes with
indegree 1
• Internal nodes
• External nodes
• Directed graph
• Complete binary tree
• Almost complete binary tree
Tree terminology(contd.)
• Ancestor of a node are all the nodes along the path from the root
to the node
• Descendants of a node are all the nodes that are in its subtrees
• Root is at level one
• Subsequent nodes have the level one level of the node’s parent
plus one
• Height or depth of the tree is the maximum level of any node in the
tree
Representation of Trees
• List Representation
• Draw the tree having the list representation of (A(B(E(K,L),F),C(G),D(H(M),I,J)))
Data
Link1
Link2
…
Linkn
• Left Child-Right Sibling Representation
Data
Left Child
Right Sibling
• Representation as a degree two tree (Left child-Right Child tree)
Terminology of binary Trees
• A binary tree is a finite set of nodes that is either empty or consists of a
root and two disjoint binary tree called the left subtree and the right
subtree.
• It is a finite set of elements that is either empty or is partitioned into three
disjoint subsets- root, left and right subtrees –both of which can be empty
• Each element of a binary tree is called a node of the tree
• Nonleaf nodes are internal nodes and leaves external
• Father, left/ right son
• Skewed tree is one in which all nodes in the tree are on to the left or to
the right
• Difference between tree and binary tree?
• Degree two
• Left and right subtree
• Tree have zero nodes
• A binary tree with n nodes and depth k is complete iff its nodes
correspond to the nodes numbered from 1 to n in the full binary
tree of depth k
Properties of binary trees
• Lemma: Maximum number of nodes
• The maximum number of nodes on level i of a binary tree is 2 i-1 , i>=1
• Proof:
• Induction base: The root is the only node on level i=1. Hence,
maximum number of nodes on level i= 1 is 2 i-1 = 2 0 =1
• Induction hypothesis:
• For all j, 1<=j<I, the maximum number of nodes on level j is 2 j-1
• Induction step: The maximum number of nodes on level i-1 is 2 i-2 by
induction hypothesis. Since each node in a binary tree has a maximum
degree of 2, the maximum number of nodes on level i is two times the
maximum number of nodes on level i-1 on 2 i-1
The maximum number of nodes in a binary tree of depth k is
𝒌
𝒌
𝒊 − 𝟏 =2k - 1 , k>=1
𝒎𝒂𝒙
𝒏𝒖𝒎𝒃𝒆𝒓
𝒐𝒇
𝒏𝒐𝒅𝒆𝒔
𝒐𝒏
𝒍𝒆𝒗𝒆𝒍
𝒊
=
𝟐
𝒊=𝟏
𝒊=𝟏
Properties of binary tree
• Lemma: Relation between number of leaf nodes and nodes of degree 2
• For any nonempty binary tree, T, if n 0 is the number of leaf nodes and n 2 the
number of nodes of degree 2, then n 0=n2+1
• Proof:
• Let n1 be the number of nodes of degree one and n the total number of nodes.
Since all in T are of degree at most two, we have
•
n= n0+n1+n2
(1)
• If we count the number of branches in a binary tree, we see that every node
except the root has a branch into it
• If B is the number of branches then n=B+1.
• All branches stem from a node of degree one or two. Thus B=n 1+2n2.
• We obtain
n=1+n 1+2n2
(2)
• Subtracting (2) from (1) and rearranging terms, we get
•
n0=n2+1
Binary tree representations
• A full binary tree of depth k is a binary tree of depth k having 2 k -1
nodes, K>=0.
• A binary tree with n nodes and depth k is complete iff its nodes
correspond to the nodes numbered from 1 to n in the full binary
tree of depth k.
• If a complete binary tree with n nodes
• (depth = log 2𝑛 + 1 is represented sequentially then for any node
with index i, 1<=i<=n, we have
• Parent(i) is at 𝑖/2 𝑖𝑓 𝑖 ≠ 1. 𝐼𝑓 𝑖 =
1, 𝑎𝑡 𝑡ℎ𝑒 𝑟𝑜𝑜𝑡 𝑎𝑛𝑑 ℎ𝑎𝑠 𝑛𝑜 𝑝𝑎𝑟𝑒𝑛𝑡
• Leftchild (i) is at 2i if 2i<=n. If 2i<n , then i has no left child
• Rightchild(i) is at 2i+1 if 2i+1<=n. If 2i+1>n, then i has no right child
Storage representation of binary trees:
• Trees can be represented using sequential allocation
techniques(arrays) or by dynamically allocating memory.
• In 2nd technique, node has 3 fields
1.
Info : which contains actual information.
2.
Llink :contains address of left subtree.
3.
Rlink :contains address of right subtree.
struct node
{
int info;
struct node *llink;
struct node *rlink;
};
typedef struct node *NODEPTR;
Expression tree
• A node representing an operator is a non leaf whereas a node
representing an operand is a leaf
• Postorder traversal places an operator after its two operands –
produces postfix form of expression
• Preorder traversal –Prefix expression
• Inorder traversal–Infix expression
Binary Search Tree
• A binary search tree is a binary tree. It may be empty. If it is not
empty, it satisfies the following properties:
• Every element has a key, and no two elements have the same key, that is,
the keys are unique
• The keys in a non empty left sub tree must be smaller than the key in the
root of the sub tree
• The keys in a non empty right sub tree must be larger than the key in the
root of the sub tree
• The left and right sub trees are also binary search tree.
Balanced trees(AVL trees) –(Adleson-Velskii and Landis)
• The balanced trees where height is used as a criterion for balancing is
height balanced trees
• An empty binary tree is height balanced
• If T is a nonempty binary tree with T L and TR as its left and right
subtrees, then T is height balanced if
• TL and TR are height balanced
• |hL –hR| <=1 where hL and hR are height of TL and TR resp.
• Weight balanced are the ones where the number of external nodes form
the basis
• Balance factor of a node, T in a binary tree is defined as hL-hR where hL
and hR are respectively, the heights of the left and right sub trees of T.
For any node T in an AVK tree BF(T)= -1, 0 or 1.