  This paper concentrates on the set $\mathcal{V}_n$ of weighted Dyck paths of
length $2n$ with special restrictions on the level of valleys. We first give
its explicit formula of the counting generating function in terms of certain
weight functions. When the weight functions are specialized, some connections
are builded between $\mathcal{V}_n$ and other classical combinatorial
structures such as $(a,b)$-Motzkin paths, $q$-Schr\"{o}der paths, Delannoy
paths and complete $k$-ary trees. Some bijections are also established between
these settings and $\mathcal{V}_n$ subject to certain special weight functions.
