We investigate the distribution of statistical measures of tree imbalance in large phylogenies. More specifically, we study normalized versions of the Sackin's index and the number of subtrees of given sizes. Using the connection with structures from theoretical computer science, we provide precise description for the limiting distribution under the null hypothesis of Yule trees. Corrected p-values are then computed, and the statistical power of these statistics for testing the Yule model against a model of biased speciation is evaluated from simulations. As an illustration, the tests are applied to the HIV-1 reconstructed phylogeny.