In this paper we introduce two new classes of $BCI$-algebras, 

namely the class of branchwise positive implicative $BCI$-algebras 

and the class of branchwise implicative $BCI$-algebras.  

The class of branchwise positive implicative $BCI$-algebras 

contains the class of positive implicative $BCK$-algebras [10], 

the class of medial $BCI$-algebras, the class of positive 

implicative $BCI$-algebras [1, 4, 13] and the class of branchwise 

implicative $BCI$-algebras contains the class of implicative 

$BCK$-algebras [10], the class of weakly implicative $BCI$-algebras [1] 

and the class of quasi-implicative $BCI$-algebras [17].  

Necessary and sufficient condition for a $BCI$-algebra to be 

a branchwise implicative $BCI$-algebra have been investigated.