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 $f_0$ the one vertex tree and $f_1$ the tree that consists of a root with two leaves attached to it; the leaf-Fibonacci tree $f_n$ of size $n\geq 2$ is the binary tree whose branches are $f_{ n-1}$ and $f_{ n-2}$. We derive a nonlinear difference equation for the number $\text{ N}(f_n)$ of nonisomorphic leaf-induced subtrees (subtrees induced by leaves) of $f_n$, and also prove that $\text{ N}(f_n)$ is asymptotic to $1.00001887227319\ldots (1.48369689570172 \ldots)^{ \phi^n}$ ($\phi=$~golden ratio) as $n$ grows to infinity.
Categories: math.CO