Consider a rooted $n$ node binary tree represented using pointers. The best upper bound on the time…

Question

Consider a rooted $n$ node binary tree represented using pointers. The best upper bound on the time required to determine the number of subtrees having exactly 4 nodes is $O(n^a\log^bn)$. Then the value of $a + 10b$ is __________.

Accepted Answer

Correct answer: 1 — Key idea: compute the size of the subtree rooted at each node using a single post-order traversal. Step 1: Perform a post-order traversal (process children before the parent). Step 2: For each node, compute subtree_size = 1 + size(left_child) + size(right_child). Store or return this size to the parent. Step 3: Whenever subtree_size equals 4, increment a counter. Time complexity: each node is visited once and processed in O(1) time (accessing children via pointers and adding sizes), so the overall time is O(n). Conclusion: the running time is O(n) = O(n^1 log^0 n), so a = 1, b = 0 and a + 10b = 1.

Consider a rooted \(n\) node binary tree represented using pointers. The best…