A complete binary tree is a special type of binary tree in data structure that has a unique property. In a complete binary tree, all levels of the tree are fully filled except possibly for the last level, which is filled from left to right.
To better understand what this means, let’s take a look at some examples:
Consider the following binary tree:
1 / \ 2 3 / \ / 4 5 6
This is not a complete binary tree because the last level is not fully filled. The node with value 6 is missing its left child.
Now let’s consider another binary tree:
1 / \ 2 3 / \ / \ 4 5 6 7
This is a complete binary tree because all levels are fully filled. Each level has the maximum number of nodes possible except for the last level, which is filled from left to right.
Properties of Complete Binary Trees:
A complete binary tree has some interesting properties:
- All levels except possibly the last level are completely filled. This means that all nodes on each level have two children, except for the last level which may be partially filled from left to right.
- All nodes in the last level are as far left as possible. This property ensures that there are no gaps between nodes in the last level. If there is any gap, then it should be towards the right side.
- The height of a complete binary tree is the minimum possible. This means that a complete binary tree with n nodes has a height of log2n.
Applications of Complete Binary Trees:
Complete binary trees have various applications in data structures and algorithms:
- Heap data structure: Heaps are often implemented using complete binary trees. In a heap, the maximum or minimum element can be efficiently extracted in O(log n) time.
- Binary heap: A binary heap is a complete binary tree that satisfies the heap property. It is used for efficient implementation of priority queues.
- Huffman coding: Huffman coding is a lossless data compression algorithm that uses complete binary trees to build optimal prefix codes for characters based on their frequencies.
In conclusion, a complete binary tree is a special type of binary tree where all levels except possibly the last level are completely filled, and all nodes in the last level are as far left as possible. Complete binary trees have applications in various data structures and algorithms, making them an important concept to understand in computer science and programming.