Binary heap
From Academic Kids

Binary heaps are a particularly simple kind of heap data structure created using a binary tree. It can be seen as a binary tree with two additional constraints:
 The shape property: the tree is either a perfect binary tree, or, if the last level of the tree is not complete, the nodes are filled from left to right. (If the nodes are numbered level by level, starting at the root with 1, this produces the useful property that the <math>{n}<math>th node of the heap is a child of the <math>\left \lfloor \frac{n}{2} \right \rfloor<math>th node.)
 The heap property: each node is greater than or equal to each of its children.
"Greater than" means according to whatever comparison function is chosen to sort the heap, not necessarily "greater than" in the mathematical sense (since the quantities are not even always numerical). Heaps where the comparison function is mathematical "greater than" are called maxheaps; those where the comparison function is mathematical "less than" are called "minheaps". Conventionally, maxheaps are used, since they are readily applicable for use in priority queues.
1 11 / \ / \ 2 3 9 10 / \ / \ / \ / \ 4 5 6 7 5 6 7 8 / \ / \ / \ / \ 8 9 10 11 1 2 3 4
Note that the ordering of siblings in a heap is not specified by the heap property, so the two children of a parent can be freely interchanged (as long as this does not violate the shape property). Compare: Treap
Contents 
Heap operations
Adding to the heap
If we have a heap, and we add an element, we can perform an operation known as upheap, bubbleup, or siftup in order to restore the heap property. We can do this in O(log n) time, using a binary heap, by adding the element on the bottom level of the heap regardless, then considering the added element and its parent and swapping the element and its parent if need be until we are assured the heap property remains. We do this at maximum for each level in the tree — the height of the tree, which is O(log n).
Say we have a maxheap
11 / \ 5 8 / \ / 3 4 X
and we want to add the number 15 to the heap. We first place the 15 in the position marked by the X. However the heap property is violated since 15 is greater than 8, so we need to swap the 15 and the 8. So, we have the heap looking as follows after the first swap:
11 / \ 5 15 / \ / 3 4 8
However the heap property is still violated since 15 is greater than 11, so we need to swap again:
15 / \ 5 11 / \ / 3 4 8
which is a valid maxheap.
Deleting the root from the heap
The procedure for deleting the root from the heap  effectively giving the maximum element in a maxheap or the minimum element in a minheap  is similar to upheap. What we do is remove the root, then replace it with the last element on the last level. So, if we have the same maxheap as before,
11 / \ 5 8 / \ 3 4
we remove the 11 and replace it with the 4.
4 / \ 5 8 / 3
Now the heap property is violated since 8 is greater than 4. If we swap these two elements, we have restored the heap property and we need not swap elements further:
8 / \ 5 4 / 3
Heap implementation
It is perfectly possible to use a traditional binary tree data structure to implement a binary heap. There is an issue with finding the adjacent element on the last level on the binary heap when adding an element which can be resolved algorithmically or by adding extra data to the nodes, called "threading" the tree — that is, instead of merely storing references to the children, we store the inorder successor of the node as well.
However, a more common approach is to store the heap in an array. Any binary tree can be stored in an array, but because a heap is always a complete binary tree, it can be stored compactly. No space is required for pointers; instead, for each index i, element a[i] is the parent of two children a[2i+1] and a[2i+2], as shown in the figure. This approach is particularly useful in the heapsort algorithm, where it allows the space in the input array to be reused to store the heap.
The upheap/downheap operations can be stated then in terms of an array as follows: suppose that the heap property holds for the indices b, b+1, ..., e. The siftdown function extends the heap property to b1, b, b+1, ..., e. Only index i = b1 can violate the heap property. Let j be the index of the largest child of a[i] within the range b, ..., e. (If no such index exists because 2i > e then the heap property holds for the newly extended range and nothing needs to be done.) By swapping the values a[i] and a[j] the heap property for position i is established. The only problem now is that the heap property might not hold for index j. The siftdown function is applied tailrecursively to index j until the heap property is established for all elements.
The siftdown function is fast. In each step it only needs two comparisons and one swap. The index value where it is working doubles in each iteration, so that at most log_{2} e steps are required.