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We consider dynamic random trees constructed using an attachment function $f: \mathbb{N} \to \mathbb{R}_+$ where, at each step of the evolution, a newvertex attaches to an existing vertex $v$ in the current tree with probabilityproportional to $f$(degree(v)). We explore the effect of a change point in thesystem; the dynamics are initially driven by a function f until the treereaches size $\tau(n) \in (0,n)$, at which point the attachment functionswitches to another function, $g$, until the tree reaches size $n$. Two changepoint time scales are considered, namely the standard model where $\tau(n) =\gamma n$, and the quick big bang model where $\tau(n) = n^\gamma$, for some $0< \gamma < 1$. In the former case, we obtain deterministic approximations forthe evolution of the empirical degree distribution (EDF) in sup-norm and usethese to devise a provably consistent non-parametric estimator for the changepoint $\gamma$. In the latter case, we show that the effect of pre-change pointdynamics asymptotically vanishes in the EDF, although this effect persists infunctionals such as the maximal degree. Our proofs rely on embedding thediscrete time tree dynamics in an associated (time) inhomogeneous continuoustime branching process (CTBP). In the course of proving the above results, wedevelop novel mathematical techniques to analyze both homogeneous andinhomogeneous CTBPs and obtain rates of convergence for functionals of suchprocesses, which are of independent interest.

Fluctuation Bounds for Continuous Time Branching Processes and Evolution of Growing Trees With a Change Point