What is the maximum number of nodes in a complete, binary tree of height 8?
511
1023
17
None of these
The correct answer and explanation is:
The correct answer is 511.
A complete binary tree is a type of binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible. However, the term “maximum number of nodes” usually refers to a perfect binary tree, which is a special case of a complete binary tree where all levels are fully filled.
In a perfect binary tree, the total number of nodes is given by the formula: Maximum nodes=2(h+1)−1\text{Maximum nodes} = 2^{(h+1)} – 1
where h is the height of the tree. Height is usually defined as the number of edges on the longest downward path from the root to a leaf, or the number of levels minus one. In most contexts in computer science, height 0 means the tree has only the root node. So if a tree has height 8, it has 9 levels (levels 0 through 8).
Using the formula: 2(8+1)−1=29−1=512−1=5112^{(8+1)} – 1 = 2^9 – 1 = 512 – 1 = 511
Therefore, the maximum number of nodes in a complete (perfect) binary tree of height 8 is 511.
This result makes sense because each level of a binary tree has twice as many nodes as the previous level. For example, level 0 has 1 node, level 1 has 2 nodes, level 2 has 4 nodes, and so on. If you add up all these nodes for levels 0 through 8, you get the same sum: 1+2+4+8+16+32+64+128+256=5111 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 = 511
This demonstrates the power of binary trees to grow exponentially with each additional level. Understanding this property is essential in computer science, as binary trees form the basis for data structures like heaps and balanced search trees.