In early 20’s,  the first person to notice the link between Random  Forests  (RF)and  Kernel  Methods, Leo  Breiman(Breiman,  2000), pointed out that  RandomForests grown using independent and identically distributed random variables in the tree construction is equivalent to kernels acting on true distribution.  Later, Scornet (Scornet, 2016b) defined Kernel-based Random Forest (KeRF) estimates and gave explicit expression for the kernels based on Centered RF and Uniform RF. In this paper, we will study the general expression for the connection function(kernel function) of an RF when splits/cuts are performed according to uniform distribution and also according to any general distribution.  We also establish the consistency of KeRF estimates in both cases and their asymptotic normality.