Define a complete binary tree and its properties.

A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled.

A binary tree is a tree data structure in which each node has at most two children, referred to as the left child and the right child. A complete binary tree is a special type of binary tree that has some unique properties. In a complete binary tree, all levels of the tree are fully filled except for the last level, which is filled from left to right. This means that if you were to look at the tree level by level, you would see that each level is completely filled with nodes until you reach the last level.

One of the key properties of a complete binary tree is its height. The height of a complete binary tree is always the smallest possible for the number of nodes in the tree. This is because the tree is filled level by level, starting from the root and moving down to the leaves. The height of a complete binary tree with 'n' nodes is log(n+1) base 2, rounded down to the nearest whole number.

Another important property of a complete binary tree is that it is always a balanced tree. A balanced tree is one in which the difference in height between the left and right subtrees of any node is at most one. This property ensures that operations such as insertion, deletion, and search can be performed efficiently on a complete binary tree.

In terms of representation, complete binary trees are typically represented using arrays. This is because the properties of a complete binary tree ensure that there are no gaps in the array representation. The left child of a node at index 'i' in the array can be found at index '2i+1', and the right child can be found at index '2i+2'. Similarly, the parent of a node at index 'i' can be found at index '(i-1)/2', rounded down.

In conclusion, a complete binary tree is a special type of binary tree that is fully filled except for the last level, which is filled from left to right. It has several important properties that make it efficient for various operations and suitable for array representation.