Leaf-induced subtrees of leaf-Fibonacci trees

Abstract

In analogy to a concept of Fibonacci trees, we define the leaf-Fibonacci tree of size n and investigate its number of nonisomorphic leaf-induced subtrees. Denote by f0 the one vertex tree and by f1 the tree that consists of a root with two leaves attached to it; the leaf-Fibonacci tree fn of size n ≥ 2 is the binary tree whose branches are fn−1 and fn−2 . We derive a nonlinear difference equation for the number N(fn ) of nonisomorphic leaf-induced subtrees (subtrees n induced by leaves) of fn , and also prove that N(fn ) is asymptotic to K1 · K2φ as n tends to infinity, where φ is the golden ratio and K1 , K2 are explicitly calculated constants.