## Saturday, June 29, 2013

### Number of nodes in a complete binary tree

How many nodes are there in a full complete binary tree of level 6?

a) 40             b) 31               c) 127               d) 128

A binary tree's level starts at 0,  a full complete binary tree has 2^i nodes at level i. At level 0 there is 1 node, at level 1 there are 2 nodes and  it goes on.

There are 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 = 2 ^ (6+1) - 1 = 2^7 - 1 = 127 nodes.

A level is also called the height of the binary tree. In a full complete binary tree of height 'h' there are 2^(h+1) - 1 nodes.

2^0+ 2^1+ 2^2 + 2^3+...................... +2^h = 2^(h+1) - 1

A full binary tree (sometimes proper binary tree or 2-tree) is a tree in which every node other than the leaves has two children. ﻿A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes in the last level are filled in left to right order.

In a complete binary tree of height 'h' there can be [2^h  2^(h+1) - 1] nodes.  A complete binary tree can have 2^h minimum number nodes or 2^(h+1)-1 maximum number of nodes.

In a complete binary tree of height 2, there can be 4 minimum number of nodes and 7 maximum number of nodes.

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