The one-node tree proportion conjecture for quadratic functional graphs

For prime $p$, let $t_p(a,k)$ be the number of binary trees with $k$ nodes attached to cyclic points in the functional graph ${\mathcal G}_a$ of $f_a(X)=X^2+a$. Define

$$T_p(k)=\sum_{a\in\mathbb F_p}t_p(a,k),\qquad T_p=\sum_{k=1}^{p-1}T_p(k).$$

The one-node tree conjecture.

$$\frac{T_p(1)}{T_p}\sim\frac12\quad\text{as }p\to\infty.$$

The conjecture is motivated by the observation that approximately half of the nonzero elements of $\mathbb F_p$ are nonsquares, which should produce a substantial proportion of trees with one node.