Sweep through all the elements to find the smallest item, then the smallest remaining item, etc. until the array is sorted. It is clear this algorithm must be correct from an inductive argument, since the i th element is in its correct position. It is clear that this algorithm takes time. More careful analysis doubles this. Thus selection sort runs in time.
All leaves are on, at most, two adjacent levels.
All leaves on the lowest level occur to the left, and all levels except the lowest one are completely filled. The key in root is all its children, and the left and right subtrees are again binary heaps. Conditions 1 and 2 specify shape of the tree, and condition 3 the labeling of the tree. Partial orders can be used to model heirarchies with incomplete information or equal-valued elements. The partial order defined by the heap structure is weaker than that of the total order, which explains Why it is easier to build.