On connected sub-tree of Cayley tree of order 2 with fixed nodes - Mukhamedov, Farrukh; Pah, Chin Hee; Saburov, Mansoor (American Institute of Physics, 2013)
We found an exact formulation for a finite sub-tree counting problem. Solution to two extremal cases are Catalan Triangle introduced by Shapiro and ballot Catalan triangles. The general solution could be expressed as linear combination of these Catalan triangles.