Inserting a new node

Inserting into a heap-tree is relatively straightforward. Because we keep of the largest position n which has been filled so far, we can insert the new element at position ${\tt n} + 1$, provided there is still room in the array. This will once again give us a complete binary tree, but the heap-tree property might, of course, be violated now. Hence we may have to `bubble up' the new element. This is done by comparing its priority with that of its parent, and if the new element has higher priority larger, then it is exchanged with its parent. We may have to repeat this process, but once we reach a parent that has bigger or equal priority, we can stop. Inserting a node takes at most $\log n$ steps because the number of times that we may have to `bubble up' the new element depends on the height of the tree. Here is an algorithm

  insert(int v) { 
    if (n < MAX) { 
       n = n + 1;
       heap[n] = v;   
       bubble_up(n);
       }
    else
       raise an exception because we've run out of space; 
  }

  bubble_up(int i) { 
    while (not isroot(i) and  heap[i] > heap[parent(i)]) { 
          swap heap[i] and heap[parent(i)];
          i = parent(i);
      }
    }