Many real-world complex networks arise as a result of a competition between growth and rewiring processes. Usually the initial part of the evolution is dominated by growth while the later one rather by rewiring. The initial growth allows the network to reach a certain size while rewiring to optimize its function and topology. As a model example we consider tree networks which first grow in a stochastic process of node attachment and then age in a stochastic process of local topology changes. The ageing is implemented as a Markov process that preserves the node-degree distribution. We quantify differences between the initial and aged network topologies and study the dynamics of the evolution. We implement two versions of the ageing dynamics. One is based on reshuffling of leaves and the other on reshuffling of branches. The latter one generates much faster ageing due to nonlocal nature of changes.