On isomorphism classes of leaf-induced subtrees in topological trees

A subtree can be induced in a natural way by a subset of leaves of a rooted tree. We study the number of nonisomorphic such subtrees induced by leaves (leaf-induced subtrees) of a rooted tree with no vertex of outdegree 1 (topological tree). We show that only stars and binary caterpillars have the minimum nonisomorphic leaf-induced subtrees among all topological trees with a given number of leaves. We obtain a closed formula and a recursive formula for the families of $d$-ary caterpillars and complete $d$-ary trees, respectively. An asymptotic formula is found for complete $d$-ary trees using polynomial recurrences. We also show that the complete binary tree of height $h>1$ contains precisely $\lfloor 2(1.24602...)^{2^h}\rfloor$ nonisomorphic leaf-induced subtrees.