In a height balanced binary search tree, the heights of the left and right descendants of any node differ by at most 1.
This is the definition of $AVL$ tree which is also known as Balanced Binary Search tree.
In this height of the tree: $O$$\left ( \log n \right )$
worst case and complexity of searching an element in $AVL$ tree: $O$$\left ( \log n \right )$
Average case complexity of searching an element in $AVL$ tree: $\Theta \left ( \log n \right )$
so option $B$