Algorithm for searching in BST with only <
If what you mean is that you want to build a BST and you only have the $&lt;$ operation, and you only know the algorithms with the $\leq$ operation, you can notice that :
$$a \leq b \Leftrightarrow \neg (a &gt; b)$$
$$\Leftrightarrow \neg(b &lt; a )$$
Hence, using a negation in the right place, you can build your usual algorithm.