Let \(h(t)\) be a continuous increasing function on \([0,1]\) with \(h(0) =0\) and \(h(t) >0\) if \(0<t\leq 1\). A compact set \(\Gamma\) on \(\mathbb R^n\) is called an \(h\)-set if there is a Radon measure \(\mu\) with \(\text{supp\,}\mu =\Gamma\) and \(\mu \big(B(\gamma, r)\big) \sim h(r)\), \(0<r<1\), where \(B(\gamma,r)\) is the ball centered at \(\gamma \in \Gamma\) and of radius \(r\) (the best known examples are \(d\)-sets, where \(h(t) = t^d\), \(0<d<n\)). Let \(B^s_{p,q} (\Gamma)\) be Besov spaces on \(\Gamma\) defined as traces of suitable Besov spaces \(B^\sigma_{p,q} (\mathbb R^n)\) of generalized smoothness \(\sigma\) on \(\Gamma\). The growth envelope function for a spaces \(X\) (on \(\mathbb R^n\) or on \(\Gamma\)) is given by \[ {\mathcal E}^X_G (t) = \sup \big\{ f^* (t): \| \;f \, | X\| \leq 1 \big\}, \qquad 0<t<1. \] The paper studies in detail properties of envelope functions for the spaces \(B^s_{p,q} (\Gamma)\). Theorem 4.6 is the main result.