# Introduction to Binary Heaps

Today, we will discuss a binary heaps, a common data structure in programming. Heaps have multiple uses in programming including:

**Priority Queues**: a queue-like data structure which has a priority field as a key. In a priority queue implemented with a min-heap, the element with the lowest value for priority will the next available element. In a priority queue implemented with a max-heap, the element with the highest value for priority will be the next available element.**Finding the k’th [small/larg]est element**: Heaps are really good at solving this problem, because after creating a heap from the target elements, the lowest (min-heap implementation) or the highest (max-heap implementation) values can be extracted with ease.

Heaps are very useful when you need to retrieve the max (max-heap) or min (min-heap) quickly (\(O(\log n)\)), need to add an element with a certain value quickly (\(O(\log n)\)), and need to be able to build the structure quickly (\(O(\log n)\)).

Binary heaps are a data structure with two important properties:

**Nearly****Complete**: every level of the tree is filled, except the last level, which can be partially filled from left to right.- In a
**min-heap**, a child cannot be smaller than its parent; in a**max-heap**, a child cannot be larger than its parent.

Because of the second property, in a max-heap, the root node is the largest value; in a max-heap, it is the smallest value.

- Below is a proper max-heap. Note that no child is greater than its parent?:
- Below is a proper min-heap. Note that no child is smaller than its parent?:
- What is wrong with the tree below?
- It is not nearly-complete. To be nearly complete, every row must be filled from top to the bottom, except the last row, which must be filled from left to right until full.
- In order for a tree to be nearly complete, every row except the last must be completely filled. The last row should be filled from left to right.
- In this example, the bottom row is not filled from left to right.

- Why is the following tree not a heap?
- This tree does not conform with the min-heap property that every child should be larger than its parent. In this case, 8 is less than 10, so it fails.