Abstract
The first and second Zagreb indices are the molecular descriptors that were appeared within a study of molecular modeling more than four decades ago. The coindex versions of these Zagreb indices were introduced around one decade before. Finding trees with the first two extremum Zagreb indices/coindices from the class of all chemical trees of a fixed order is a chemical-graph-theoretic problem whose partial solution is known. This paper is devoted to reporting the complete solution of the aforementioned problem.