Definition:
A binary tree is a non-linear tree data structure whose all
nodes have either zero, one, or at most two children nodes. These two
children are generally referred to as left and right child respectively.
Properties:
At each level of i, the maximum number of nodes is 2i.
○
If a binary tree has L number of leaf nodes, its height is given by L +
1.
○
The height of the tree is defined as the longest path from the root
node to the leaf node. Say a tree has height equal to 3. Therefore, the
maximum number of nodes at height 3 is equal to (1+2+4+8) = 15.
In general, the maximum number of nodes possible at height h is (20
+ 21 + 22+….2h) = 2h+1
-
1.
○
The minimum number of nodes possible at height h is equal to h+1.
○
If the number of nodes is minimum, then the height of the tree would
be maximum. Conversely, if the number of nodes is maximum, then
the height of the tree would be minimum.
○
If there are 'n' number of nodes in the binary tree.
The minimum height can be computed as:
As we know that,
n = 2h+1
-
1
n+1 = 2h+1
Taking log on both the sides,
log2(n+1) = log2(2h+1)
log2(n+1) = h+1
h = log2(n+1)
-
1
The maximum height can be computed as:
As we know that,
n = h+1
h= n
-
1
Types of binary tree:
Full/proper/strict binary tree:
The full binary tree is also known
as a strict binary tree. The tree can only be considered as the full
binary tree if each node must contain either 0 or 2 child node of its
parent node or the parent node is itself the leaf node or the external
node.
In other words a full binary tree is either a single vertex and the
root node has two subtrees, both of which are full binary trees.
Here, the quantity of leaf nodes is equal to the number of internal
nodes plus one.
○
In the above tree, we can observe that each node is either containing zero
or two children; therefore, it is a Full Binary tree.
Properties of Full Binary Tree
The number of leaf nodes is = the number internal nodes(i.e.,
excluding root and leaf nodes) + 1.
▪
The maximum number of nodes is the same as the number of
nodes in the binary tree, i.e., 2h+1
-
1.
▪
The minimum number of nodes in the full binary tree is 2*h
-
1.
▪
The minimum height of the full binary tree is log2(n+1)
-
1.
▪
The maximum height of the full binary tree can be computed as:
n= 2*h
-
1
n+1 = 2*h
h = n+1/2
▪
►
Complete binary tree:
The complete binary tree is a tree in which
all the nodes are completely filled except the last level that may or may
○
Array implementation of Binary tree:
CASE I:
If a node is at ith index:
Left child would be at: [(2 * i) + 1]
▪
Right child would be at: [(2 * i) + 2]
▪
Parent would be at:
▪
For example,
For D,
Left child = [(2 * 3) + 1] = 7 (H)
Right child = [(2 * 3) + 2] = 8 (I)
Parent =
(B)
For E,
Left child = [(2 * 4) + 1] = 9 (As there's no child of E)
Same for right child.
CASE II:
If a node is at ith index:
Left child would be at: [(2 * i)]
▪
Right child would be at: [(2 * i) + 1]
▪
Parent would be at:
▪
For example,
For D,
Left child = [(2 * 4)] = 8 (H)
Right child = [(2 * 4) + 1] = 9 (I)
Parent =
= 2(B)
For E,
Left child = [(2 * 5)] = 10 (As there's no child of E)
Same for right child.
For finding children using these formulas the tree have to be a Complete
Binary Tree.
In case of the tree is not complete we have to make it complete by
placing empty nodes. Here if D does not have any child and E have so
to make it complete we have to place empty nodes in left.
In that case the array representation will be like this:
For example,
For E,
Left child = [(2 * 4) + 1] = 9 (H)
Right child = [(2 * 4) + 2] = 10 (I)
Parent =
(B)
BINARY TREE
09 December 2022
17:45
DAA Page 1
