We study a random tree model known as the Preferential Attachment tree with Vertex Death. Here, one can both add vertices to the tree as well as kill vertices. This model mimics the non-monotone growth of real-world networks, absent in classical preferential attachment models. One initialises the tree with a single root vertex labelled 
1. At every step n, either a new vertex labelled n+1 is added to the tree and connected to an already present alive vertex selected preferentially according to a function b, or an already present vertex is selected preferentially according to a function d  and killed. Killed vertices can make no new connections. We are interested in the behaviour of the richest alive vertex I_n (with the largest degree) and the oldest alive vertex O_n (with the smallest label). When I_n converges almost surely, we say that a persistent hub exists. When I_n  does not converge but I_n/O_n is tight, we say that persistence occurs, and when I_n/O_n diverges to infinity we say lack of persistence occurs. We uncover three distinct regimes in which behaviour is different: (1) The infinite lifetime regime, where we provide conditions under which a persistent hub exists almost surely. (2) The rich are old regime, in which O_n does not converge and where we provide conditions under which either persistence or lack of persistence occurs. (3) The rich die young regime, in which O_n does not converge and where lack of persistence always occurs. We shall discuss how the three regimes can be identified and what drives the behaviour observed in each regime. Partly joint work with Markus Heydenreich.