Recently Chen-Liang-Wang (Linear Algerbra Appl. 471 (2015) 383—393) proved some sufficient conditions for the total positivity of Catalan-Stieltjes matrices. Our aim is to provide a combinatorial interpretation of their sufficiant conditions. More precisely, for any Catalan-Stieltjes matrix $A$ we construct a digraph with a weight, which is positive under their sufficient conditions, such that every minor of $A$ is equal to the sum of weights of families of nonintersecting paths of the digraph. We have also an analogue result for the minors of Hankel matrix associated to the first column of Catalan-Stieltjes matrix $A$.