Copper-catalyzed azide-alkyne cycloaddition polymerization (CuAACP) of AB(2) monomers demonstrated a chain-growth mechanism without any external ligand because of the complexation of in situ formed triazole groups with Cu catalysts. In this study, we explored the use of various ligands that affected the polymerization kinetics to tune the polymers' molecular weights and the degree of branching (DB). Eight ligands were studied, including polyethylene glycol monomethyl ether (PEG(350), M-n = 350), tris(benzyltriazolylmethyl)amine (TBTA), 2,6-bis(1-undecyl-1H-benzo[d]imidazol-2-yl)pyridine (Py(DBim)(2)), 2,2-bipyridyl (bpy), 4,4-di-n-nonyl-2,2-bipyridine (dNbpy), N,N,N,N,N-pentamethyldiethylenetriamine (PMDETA), N,N,N,N,N-penta(n-butyl)diethylenetriamine (PBuDETA), and N,N,N,N,N-pentabenzyldiethylenetriamine (PBnDETA). All ligands except PEG(350) exhibited stronger coordination with Cu(I) than the polytriazole polymer, which freed the Cu catalyst from polymers and resulted in dominant step-growth polymerization with simultaneous chain-growth feature. Meanwhile, the use of PEG(350) ligand retained the confined Cu in the polymer, demonstrating a chain-growth mechanism, but lower polymer molecular weights as compared with the no-external-ligand polymerization. Results indicated that aliphatic substituent groups on ligands had little effect on the molecular weights and DB of the polymers, but rigid aromatic substituent groups decreased both values. By varying the ligand species and amounts, hyperbranched polymers with DB value ranging from 0.53 ([TBTA](0)/[Cu](0) = 5) to 0.98 ([PMDETA](0)/[Cu](0) = 2) have been achieved. (c) 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2018, 56, 2238-2244