We discussed an implementation of a min priority queue as a min heap, which had two key properties.

$1.$ The min heap property: the key at a node is at most the keys of its children.

$2.$ The heap is represented by a complete binary tree.

What is the purpose of the complete binary tree property? Select all that apply

$1.$ It ensures that the minimum key is at the root of the tree.

$2.$ It allows a simple representation of the tree by a vector in a memory efficient manner

$3.$ It permits us to tell if a given key is contained in the heap in time O$($log n$).$

$4.$ When the heap has n elements it insures that the height of the tree is at most log n$,$ which in turn means push and pop can be done in time O$($log n$).$