The concept of `adjunct' operation of two lattices with respect to a pair of elements is introduced. A structure theorem namely,
`A finite lattice is dismantlable if and only if it is an adjunct of chains' is obtained. Further it is established that for
any adjunct representation of a dismantlable lattice the number of chains as well as the number of times a pair of elements
occurs remains the same. If a dismantlable lattice L has n elements and n+k edges then it is proved that the number of irreducible elements of L lies between n-2k-2 and n-2. These results are used to enumerate the class of lattices with exactly two reducible elements, the class of lattices with
n elements and upto n+1 edges, and their subclasses of distributive lattices and modular lattices.