A complete tree is a type of binary tree where all levels, except possibly the last one, are completely filled with nodes. In other words, a complete tree is a binary tree in which each level is filled from left to right.
Properties of Complete Trees:
- Full Binary Tree: A complete tree is also referred to as a full binary tree because all nodes have either 0 or 2 children.
- Perfect Binary Tree: If all levels of a complete binary tree are completely filled with nodes, then it is called a perfect binary tree.
Characteristics of Complete Trees:
- Left Child Property: In a complete binary tree, if a node has a left child, then it must have a right child as well.
- Filling Order: Nodes are inserted into the complete tree from left to right, starting from the top level and moving downwards.
Determining Completeness in Data Structures:
In data structures, determining whether a binary tree is complete can be done using various algorithms. One common approach is to perform a level order traversal and check for any gaps in the sequence of nodes. If there are no gaps and all nodes are visited in the traversal, then the binary tree is considered complete.
isComplete(root): if root is null: return true queue = createQueue() queue.enqueue(root) while queue is not empty: node = queue.dequeue() if node.left: queue.enqueue(node.left) if node.right: queue.right) if node.left is null and node.right is not null: return false if node.left is null or node.right is null: while queue is not empty: temp = queue.dequeue() if temp.left or temp.right: return false return true return true
Applications of Complete Trees:
Complete trees have various applications in computer science and data structures. Some of the common applications include:
- Heap Data Structure: Heaps are often implemented as complete binary trees due to their efficient insertion and deletion properties.
- Binary Heap Sort: Complete trees are used as the underlying data structure for efficient sorting algorithms like heap sort.
- Huffman Coding: Huffman coding, a widely used compression algorithm, uses complete binary trees for encoding characters based on their frequency.
In conclusion, complete trees play a significant role in various data structures and algorithms. Understanding their properties and characteristics can greatly enhance your understanding of binary trees and their applications.