We study the randomm-ary search tree model (wherem stands for the number of branches of the search tree), an important problem for data storage in computer science, using a variety of statistical physics techniques that allow us to obtain exact asymptotic results. In particular, we show that the probability distributions of extreme observables associated with a random search tree such as the height and the balanced height of a tree have a travelling front structure. In addition, the variance of the number of nodes needed to store a data string of a given sizeN is shown to undergo a striking phase transition at a critical value of the branching ratiom_c = 26. We identified the mechanism of this phase transition and showed that it is generic and occurs in various other problems as well. New results are obtained when each element of the data string is a D-dimensional vector. We show that this problem also has a phase transition at a critical dimension,D_c = π/ sin^-1 (l/√8) = 8.69363 …