Binary Tree | Set 3 (Types of Binary Tree)

We have discussed Introduction to Binary Tree in set 1 and Properties of Binary Tree in Set 2. In this post, common types of Binary Trees are discussed.

The following are common types of Binary Trees.

Full Binary Tree A Binary Tree is a full binary tree if every node has 0 or 2 children. The following are the examples of a full binary tree. We can also say a full binary tree is a binary tree in which all nodes except leaf nodes have two children.

               18
           /       \  
         15         30  
        /  \        /  \
      40    50    100   40

             18
           /    \   
         15     20    
        /  \       
      40    50   
    /   \
   30   50

               18
            /     \  
          40       30  
                   /  \
                 100   40


In a Full Binary Tree, number of leaf nodes is the 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 a complete Binary Tree if all the levels are completely filled except possibly the last level and the last level has all keys as left as possible



The following are examples of Complete Binary Trees

               18
           /       \  
         15         30  
        /  \        /  \
      40    50    100   40


               18
           /       \  
         15         30  
        /  \        /  \
      40    50    100   40
     /  \   /
    8   7  9 

Practical example of Complete Binary Tree is Binary Heap.



Perfect Binary Tree A Binary tree is a Perfect Binary Tree in which all the internal nodes have two children and all leaf nodes are at the same level.
The following are the examples of Perfect Binary Trees.

               18
           /       \  
         15         30  
        /  \        /  \
      40    50    100   40


               18
           /       \  
         15         30  

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

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



Balanced Binary Tree
A binary tree is balanced if the height of the tree is O(Log n) where n is the number of nodes. For Example, the AVL tree maintains O(Log n) height by making sure that the difference between the heights of the left and right subtrees is almost 1. Red-Black trees maintain O(Log n) height by making sure that the number of Black nodes on every root to leaf paths is the 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.

      10
      /
    20
     \
     30
      \
      40     

Source:
https://en.wikipedia.org/wiki/Binary_tree#Types_of_binary_trees

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