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Data Structures

## Types of Binary Tree

So far we have discussed Introduction to Binary Tree and Properties of Binary Tree. In this section, common types of binary tree are discussed.

Following are common types of Binary Trees.

Full Binary Tree A Binary Tree is full if every node has 0 or 2 children. Following are examples of full binary tree.


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15         30

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40    50    100   40

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15     20

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40    50

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30   50

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100   40



In a Full Binary, number of leaf nodes is number of internal nodes plus 1
L = I + 1
Where L = Number of leaf nodes, I = Number of internal nodes
See Handshaking Lemma and Tree for proof.

Complete Binary Tree: A Binary Tree is complete Binary Tree if all levels are completely filled except possibly the last level and the last level has all keys as left as possible

Following are examples of Complete Binary Trees


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15         30

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/  \   /

8   7  9



Practical example of Complete Binary Tree is Binary Heap.

Perfect Binary Tree A Binary tree is Perfect Binary Tree in which all internal nodes have two children and all leaves are at same level.
Following are examples of Perfect Binaryr Trees.


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15         30



A Perfect Binary Tree of height h (where height is number of nodes on path from root to leaf) has 2h – 1 node.

Example of Perfect binary tree is ancestors in family. Keep a person at root, parents as children, parents of parents as their children.

Balanced Binary Tree
A binary tree is balanced if height of the tree is O(Log n) where n is number of nodes. For Example, AVL tree maintain O(Log n) height by making sure that the difference between heights of left and right subtrees is 1. Red-Black trees maintain O(Log n) height by making sure that the number of Black nodes on every root to leaf paths are same and there are no adjacent red nodes. Balanced Binary Search trees are performance wise good as they provide O(log n) time for search, insert and delete.

A degenerate (or pathological) tree A Tree where every internal node has one child. Such trees are performance-wise same as linked list.


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