Fractions (left) and rates (right) of hierarchical mergers as a function of the BBH merger rates (top) and retention probabilities (bottom) in GCs (red) and NSCs (black). The derived values of the fractions and rates are obtained according to Eqs. (4) and (5) (star symbols), respectively. The approximate values were calculated using Eqs. (7) and (6) (dashed line). The fractions and rates are plotted as a function of BBH merger rates (retention probabilities) with fixed retention probabilities of 4% and 40% (BBH merger rates fixed to 10 Gpc⁻³ yr⁻¹ and 1 Gpc⁻³ yr⁻¹) for GCs and NSCs, respectively. N_cl ∼ 1 Mpc⁻³ and ∼0.01 Mpc⁻³ and N_BH ∼ 2000 and ∼2 × 10⁴ are adopted for GCs and NSCs, respectively.