Answer
Yes
Work Step by Step
A polynomial problem is one that can be solved in polynomial time with respect to the size of the input (usually denoted as $n$).
Binary Tree Search Complexity:
In a general binary tree (not necessarily balanced or ordered), searching for a value requires visiting each node in the worst case.
If there are $n$ nodes, the worst-case time complexity is $O(n)$.
$O(n)$ is a linear time complexity, which is a subset of polynomial time $(O(n^k)$ for some constant $k$).
Special Case: Binary Search Tree (BST):
If the binary tree is a binary search tree and balanced, search can be done in $O(\log n)$ time.
$O(\log n)$ is even better than polynomial time.
Conclusion
Searching through a finite binary tree for a particular value is a polynomial problem because its worst-case time complexity is $O(n)$, which is polynomial in the size of the input.
