Let $X$ be a compact, geodesic metric space which is Gromov hyperbolic with a constant $\delta>0$. To fix scaling, let us also assume that $X$ has diameter $1$.
To fix a definition of Gromov hyperbolicity, I will use the first definition here under ``Definitions using triangles''.
I understand that mostly one does not care about the precise value of $\delta$ because we think of this as an asymptotic notion, and that all compact metric spaces are Gromov hyperbolic for $\delta$ larger than their diameters. Nonetheless, in this question I would like to care about the size of $\delta$.
Gromov hyperbolicity is often motivated as saying that a metric space looks ``tree-like'' (at least at large scales), since trees are Gromov hyperbolic with $\delta=0$. So:
Question 1: Given $\epsilon>0$, is there $\delta>0$ such that if $X$ is $\delta$-hyperbolic, then $X$ is within Gromov-Hausdorff distance $\epsilon$ from a compact metric tree?
Here a metric tree means a geodesic metric space in which each pair of points is joined by a unique arc.
I am nervous that someone might answer the above question, if the answer is indeed yes, by an appeal to ultralimits. Although this would be fine for question 1, I would be happier with a more constructive understanding. So I also ask:
Question 2: If the answer to question 1 is yes, is there an explicit estimate for the rate of decay for $\delta$ given $\epsilon$? Equivalently, is there an upper bound, tending to zero with $\delta$, for the maximal Gromov-Hausdorff distance to a tree among all metric spaces $X$ with the above properties?
I apologize if these questions are simple or well-known. If the latter, a reference would be great.