On minimal universal trees

In this paper we solve the problem of finding a minimal $n$-universal rooted tree. We show that the number $\alpha(n)$ of vertices of a minimal $n$-universal rooted tree coincides with the quantity of trees of a special form (uniform trees), the number of whose vertices $\leqslant n$. We derive a recursion formula for computing the value of $\alpha(n)$. We also specify the construction of a minimal universal tree for an arbitrary set of uniform trees.