Our goal in this paper is to determine whether the fractal dimensions (FD) of random trees is finite. Two types of random trees are considered. We first consider spherically symmetric random trees in which all vertices at level n have degree 3 with probability qn = 1/n^σ and degree 2 with probability 1 - qn. We show that FD for these trees is finite if and only if . Secondly, we consider random trees which correspond branching processes in varying environments and a similar result is obtained. Whenever we obtained a finite FD, an upper bound was found.