Many 2D covalent polymers synthesized as single layers on surfaces show inherent disorder, expressed for example in their ring-size distribution. Systems which are expected to form the thermodynamically favored hexagonal lattice usually deviate from crystallinity and include high numbers of pentagons, heptagons, and rings of other sizes. The amorphous structure of two different covalent polymers in real space using scanning tunneling microscopy is investigated. Molecular dynamics simulations are employed to extract additional information. We show that short-range correlations exist in the structure of one polymer, i. e. that polygons are not tessellating the surface randomly but that ring neighborhoods have preferential compositions. The correlation is dictated by the energy of formation of the ring neighborhoods.