The Sombor index and reduced Sombor index, introduced by mathematical chemist Ivan Gutman [MATCH Commun. Math. Comput. Chem. 86 (2021) 11–16], are the recently proposed degree-based graph invariants that attained a lot of attention from researchers in a very short time. In this paper, the best possible upper bounds on the both aforementioned indices for molecular trees are obtained in terms of order and number of branching vertices or vertices of degree 2. The optimal molecular trees achieving the obtained bounds are also completely characterized.