On connected sub-tree with fixed nodes in Cayley tree of Order 2 - Mukhamedov, Farrukh; Pah, Chin Hee; Saburov, Mansoor (Southeast Asian Mathematical Society, 2017)
In this paper we found an exact formula for a finite sub-tree counting problem. Note that the formulas, which correspond to two extremal cases, are Catalan Triangle introduced by Shapiro and ballot Catalan triangles. The general formula could be expressed as a linear combination of these Catalan triangles.