Size of the universe for van Emde Boas Trees
This assumption makes the analysis easier. If the universe $\mathcal{U}$ is of size $2^{2^k}$ then every element in $\mathcal{U}$ can be represented by $2^k$ bits. Roughly speaking, when executing one of the vEB-tree operations you halve the "relevant bits" in every level of the recursion. So when you start with $2^k$ bits, then the recursion depth is $k$.

If your universe has a size not of the form $2^{2^k}$, then you have to use floor/ceiling in the analysis.