Cogrowth formula for non-negative functions on biregular trees

Let $2\le c\le d$ and $3\le d$. Let $\mathbb{T}_{c,d}$ be the $(c,d)$-biregular tree, and let $\alpha(f)$ and $\beta(f)$ denote the quantities associated with a function $f$ on its vertices. Define

$$\overline{c}=c-1,\qquad \overline{d}=d-1.$$

For $\alpha\ge 0$, define

$$g_{c,d}(\alpha)=\begin{cases} \sqrt{\overline{c}}+\sqrt{\overline{d}} & \text{if }\alpha\le (\overline{c}\,\overline{d})^{1/4},\\ \sqrt{\alpha^2+\overline{c}+\overline{d}+\dfrac{\overline{c}\,\overline{d}}{\alpha^2}} & \text{if }\alpha\ge (\overline{c}\,\overline{d})^{1/4}. \end{cases}$$

Cogrowth formula for non-negative functions. Fix $2\le c\le d$ and $3\le d$, and let $0\ne f\colon V(\mathbb{T}_{c,d})\to\mathbb{R}_{\ge 0}$ be a non-negative function on the vertices of the $(c,d)$-biregular tree. Then

$$\beta(f)=g_{c,d}(\alpha(f)).$$

The surrounding discussion presents this as a proposed extension of the subset formula to non-negative functions; the corresponding result for arbitrary subsets is proved in the paper, but the authors state that they were unable to establish this functional generalization.