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 $O(n^a \log^b n)$. Then the value of $a + 10b$ is _________.

To find the number of subtrees having exactly 4 nodes in an $n$-node binary tree:

1. We can perform a post-order (bottom-up) traversal of the tree.

2. For each node, the size of the subtree rooted at that node can be computed in $O(1)$ time:

   $$\text{size}(u) = 1 + \text{size}(\text{left}(u)) + \text{size}(\text{right}(u))$$

3. Whenever $\text{size}(u) == 4$, we increment a count.

4. Since post-order traversal visits each node in the binary tree exactly once and performs constant $O(1)$ work per node, the overall time complexity is $O(n)$.

Comparing $O(n)$ with $O(n^a \log^b n)$:

Thus: