The logarithm–rooted-tree degree asymptotic

Let $p$ be a prime, let $t_p$ be the rooted tree corresponding to $p$, and let $\deg(t_p)$ denote its degree, namely the number of nodes in $t_p$. The logarithm of $p$ is compared with this degree by

$$\ln(p) \sim \deg(t_p).$$

Logarithm–rooted-tree degree conjecture. The degree of the rooted tree associated with a prime $p$ is asymptotic to $\ln(p)$. This is proposed as a natural measure on the linearized rational numbers related to the measure $\ln(p)\,dp$ used in the discussion of the prime number theorem. The source provides no resolution or further conditions for this asymptotic claim.