1. The Bubble-Down operation (also known as Sift-Down) for a node of height $h$ in a heap involves comparing and possibly swapping the node with its children down the tree until the heap property is restored. Since the worst-case path length from the node to a leaf is $h$, the asymptotic running time is $O(h)$ for a single Bubble-Down. 2. When performing Heapify on a binary heap with $n$ nodes, we typically call Bubble-Down on all nodes starting from the lowest level and moving upwards. 3. The total time to Bubble-Down all nodes at height $h$ corresponds to the number of nodes at height $h$ multiplied by the cost $O(h)$ per node. 4. A complete binary heap's nodes at height $h$ are roughly $\frac{n}{2^{h+1}}$. 5. Thus, the total time at height $h$ is $$O\left(h \times \frac{n}{2^{h+1}}\right).$$ 6. Summing over all heights from $0$ to $\log n$ gives total Heapify time: $$\sum_{h=0}^{\log n} O\left(h \times \frac{n}{2^{h+1}}\right) = O(n)$$ because the weighted sum converges to a constant factor times $n$. 7. Therefore, the total time complexity of Heapify is $O(n)$. Justification: Individually, Bubble-Down takes $O(h)$ time at height $h$, but fewer nodes exist at greater heights, and their decreasing count balances the increasing cost, resulting in linear total time.