  We consider models of growing random trees $\{\mathcal{T}_f(n):n\geq 1\}$
with model dynamics driven by an attachment function $f:\mathbb{Z}_+\to
\mathbb{R}_+$. At each stage a new vertex enters the system and connects to a
vertex $v$ in the current tree with probability proportional to
$f(\text{degree}(v))$. The main goal of this study is to understand the
performance of root finding algorithms. A large body of work (e.g. the work of
Bubeck, Devroye and Lugosi or Jog and Loh) has emerged in the last few years in
using techniques based on the Jordan centrality measure and its variants to
develop root finding algorithms. Given an unlabelled unrooted tree, one
computes the Jordan centrality for each vertex in the tree and for a fixed
budget $K$ outputs the optimal $K$ vertices (as measured by Jordan centrality).
Under general conditions on the attachment function $f$, we derive necessary
and sufficient bounds on the budget $K(\epsilon)$ in order to recover the root
with probability at least $1-\epsilon$. For canonical examples such as linear
preferential attachment and uniform attachment, these general results give
matching upper and lower bounds for the budget. We also prove persistence of
the optimal $K$ Jordan centers for any $K$, i.e. the existence of an almost
surely finite random time $n^*$ such that for $n \geq n^*$ the identity of the
$K$-optimal Jordan centers in $\{\mathcal{T}_f(n):n\geq n^*\}$ does not change,
thus describing robustness properties of this measure. Key technical
ingredients in the proofs of independent interest include sufficient conditions
for the existence of exponential moments for limits of (appropriately
normalized) continuous time branching processes within which the models
$\{\mathcal{T}_f(n):n\geq n^*\}$ can be embedded, as well as rates of
convergence results to these limits.
